71 The Specht Share Package

This package contains functions for computing the decomposition matrices for Iwahori--Hecke algebras of the symmetric groups. As the (modular) representation theory of these algebras closely resembles that of the (modular) representation theory of the symmetric groups --- indeed, the later is a special case of the former --- many of the combinatorial tools from the representation theory of the symmetric group are included in the package.

These programs grew out of the attempts by Gordon James and myself [JM1] to understand the decomposition matrices of Hecke algebras of type A when <q>=-1. The package is now much more general and its highlights include:

1:
Specht provides a means of working in the Grothendieck ring of a Hecke algebra H using the three natural bases corresponding to the Specht modules, projective indecomposable modules, and simple modules.

2:
For Hecke algebras defined over fields of characteristic zero we have implemented the algorithm of Lascoux, Leclerc, and Thibon [LLT] for computing decomposition numbers and ``crystallized decomposition matrices''. In principle, this gives all of the decomposition matrices of Hecke algebras defined over fields of characteristic zero.

3:
We provide a way of inducing and restricting modules. In addition, it is possible to ``induce'' decomposition matrices; this function is quite effective in calculating the decomposition matrices of Hecke algebras for small n.

4:
The q--analogue of Schaper's theorem [JM2] is included, as is Kleshchev's [K] algorithm of calculating the Mullineux map. Both are used extensively when inducing decomposition matrices.

5:
Specht can be used to compute the decomposition numbers of q--Schur algebras (and the general linear groups), although there is less direct support for these algebras. The decomposition matrices for the q--Schur algebras defined over fields of characteristic zero for n<11 and all e are included in Specht.

6:
The Littlewood--Richard rule, its inverse, and functions for many of the standard operations on partitions (such as calculating cores, quotients, and adding and removing hooks), are included.

7:
The decomposition matrices for the symmetric groups Sym_n are included for n<15 and for all primes.

The modular representation theory of Hecke algebras

The ``modular'' representation theory of the Iwahori--Hecke algebras of type A was pioneered by Dipper and James [DJ1,DJ2]; here we briefly outline the theory, referring the reader to the references for details. The definition of the Hecke algebra can be found in Chapter~Iwahori-Hecke algebras; see also Hecke.

Given a commutative integral domain R and a non--zero unit q in R, let <H>=<H>_{<R>, <q>} be the Hecke algebra of the symmetric group Sym_n on n symbols defined over R and with parameter q. For each partition mu of n, Dipper and James defined a Specht module S(mu). Let rad S(mu) be the radical of S(mu) and define D(mu)=S(mu)/rad S(mu). When R is a field, D(mu) is either zero or absolutely irreducible. Henceforth, we will always assume that R is a field.

Given a non--negative integer i, let [i]_q=1+q+ldots+q^{i-1}. Define e to be the smallest non--negative integer such that [<e>]_q=0; if no such integer exists, we set e equal to 0. Many of the functions in this package depend upon e; the integer e is the Hecke algebras analogue of the characteristic of the field in the modular representation theory of finite groups.

A partition mu=(mu_1,mu_2,ldots) is e--singular if there exists an integer i such that mu_i=mu_{i+1}=cdots=mu_{i+<e>-1}>0; otherwise, mu is e--regular. Dipper and James [DJ1] showed that D(nu)ne(0) if and only if nu is e--regular and that the D(nu) give a complete set of non--isomorphic irreducible H--modules as nu runs over the e--regular partitions of n. Further, S(mu) and S(nu) belong to the same block if and only if mu and nu have the same e-core [DJ2,JM2]. Note that these results depend only on e and not directly on R or q.

Given two partitions mu and nu, where nu is e--regular, let d_{munu} be the composition multiplicity of D(nu) in S(mu). The matrix D=(d_{munu}) is the decomposition matrix of H. When the rows and columns are ordered in a way compatible with dominance, D is lower unitriangular.

The indecomposable H-modules P(nu) are indexed by e-regular partitions nu. By general arguments, P(nu) has the same composition factors as sum_{mu} d_{munu} 'S'(<mu>) ; so these linear combinations of modules become identified in the Grothendieck ring of H. Similarly, 'D'(<nu>) = sum_{mu} d_{numu}^{-1} 'S'(<mu>) in the Grothendieck ring. These observations are the basis for many of the computations in Specht.

medskip

Two small examples

Because of the algorithm of [LLT], in principle, all of decomposition matrices for all Hecke algebras defined over fields of characteristic zero are known and available using Specht. The algorithm is recursive; however, it is quite quick and, as with a car, you need never look at the engine:

gap> H:=Specht(4);   # e=4, 'R' a field of characteristic 0
Specht(e=4, S(), P(), D(), Pq())
gap> InducedModule(H.P(12,2));
P(13,2)+P(12,3)+P(12,2,1)+P(10,3,2)+P(9,6)

The [LLT] algorithm was applied 24 times during this calculation.

For Hecke algebras defined over fields of positive characteristic the major tool provided by Specht, apart from the decomposition matrices contained in the libraries, is a way of ``inducing'' decomposition matrices. This makes it fairly easy to calculate the associated decomposition matrices for ``small'' n. For example, the Specht libraries contain the decomposition matrices for the symmetric groups Sym_n over fields of characteristic 3 for n<15. These matrices were calculated by Specht using the following commands:

gap> H:=Specht(3,3);   # e=3, 'R' field of characteristic 3
Specht(e=3, p=3, S(), P(), D())
gap> d:=DecompositionMatrix(H,5);  # known for $n\<2e$
5    
|
 1         
4,1  
|
 . 1       
3,2  
|
 . 1 1     
3,1^2
|
 . . . 1   
2^2,1
|
 1 . . . 1
2,1^3
|
 . . . . 1
1^5  
|
 . . 1 . . 
gap> for n in [6..14] do
>       d:=InducedDecompositionMatrix(d); SaveDecompositionMatrix(d);  
>    od;

The function InducedDecompositionMatrix contains almost every trick that I know for computing decomposition matrices. I would be very happy to hear of any improvements.

Specht can also be used to calculate the decomposition numbers of the q--Schur algebras; although, as yet, here no additional routines for calculating the projective indecomposables indexed by e--singular partitions. Such routines will probably be included in a future release, together with the (conjectural) algorithm [LT] for computing the decomposition matrices of the q--Schur algebras over fields of characteristic zero.

In the next release of Specht, I will also include functions for computing the decomposition matrices of Hecke algebras of type B, and more generally those of the Ariki--Koike algebras. As with the Hecke algebra of type A, there is an algorithm for computing the decomposition matrices of these algebras when R is a field of characteristic zero [M].

medskip

Credits

I would like to thank Gordon James, Johannes Lipp, and Klaus Lux for their comments and suggestions.

If you find Specht useful please let me know. I would also appreciate hearing any suggestions, comments, or improvements. In addition, if Specht does play a significant role in your research, please send me a copy of the paper(s) and please cite Specht in your references.

Andrew Mathas (Supported in part by SERC grant GR/J37690)
a.mathas@ic.ac.uk
Imperial College, 1996.

medskip

References

[A] S. Ariki, On the decomposition numbers of the Hecke algebra of G(m,1,n), preprint~(1996).

[B] J. Brundan, Modular branching rules for quantum GL_n and the Hecke algebra of type A, preprint 1996.

[DJ1] R. Dipper and G. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3), 52 (1986), 20--52.

[DJ2] R. Dipper and G. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3), 54 (1987), 57--82.

[G] M. Geck, Brauer trees of Hecke algebras, Comm. Alg., 20 (1992), 2937--2973.

[Gr] I. Grojnowski, Affine Hecke algebras (and affine quantum GL_n) at roots of unity, IMRN~5 (1994), 215--217.

[J] G. James, The decomposition matrices of GL_n(q) for n le 10, Proc. London Math. Soc.,~60 (1990), 225--264.

[JK] G. James and A. Kerber, The representation theory of the symmetric group, 16, Encyclopedia of Mathematics, Addison--Wesley, Massachusetts~(1981).

[JM1] G. James and A. Mathas, Hecke algebras of type A at q=-1, J. Algebra (to appear).

[JM2] G. James and A. Mathas, A q--analogue of the Jantzen--Schaper Theorem, Proc. London Math. Soc. (to appear).

[K] A. Kleshchev, Branching rules for modular representations III, J. London Math. Soc. (to appear).

[LLT] A. Lascoux, B. Leclerc, and J-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. (to appear).

[LT] B. Leclerc and J-Y. Thibon, Canonical bases and q--deformed Fock spaces, Int. Research Notices (to appear).

[M] A. Mathas, Canonical bases and the decomposition matrices of Ariki--Koike algebras, preprint~1996.

Subsections

  1. Specht
  2. Hecke algebras over fields of positive characteristic
  3. The Fock space and Hecke algebras over fields of characteristic zero
  4. Schur
  5. DecompositionMatrix
  6. CrystalDecompositionMatrix
  7. DecompositionNumber
  8. Partitions in Specht
  9. Inducing and restricting modules
  10. InducedModule
  11. SInducedModule
  12. RestrictedModule
  13. SRestrictedModule
  14. Operations on decomposition matrices
  15. InducedDecompositionMatrix
  16. IsNewIndecomposable
  17. InvertDecompositionMatrix
  18. AdjustmentMatrix
  19. SaveDecompositionMatrix
  20. CalculateDecompositionMatrix
  21. MatrixDecompositionMatrix
  22. DecompositionMatrixMatrix
  23. AddIndecomposable
  24. RemoveIndecomposable
  25. MissingIndecomposables
  26. Calculating dimensions
  27. SimpleDimension
  28. SpechtDimension
  29. Combinatorics on Young diagrams
  30. Schaper
  31. IsSimpleModule
  32. Mullineux
  33. GoodNodes
  34. GoodNodeSequence
  35. PartitionGoodNodeSequence
  36. GoodNodeLatticePath
  37. LittlewoodRichardsonRule
  38. InverseLittlewoodRichardsonRule
  39. EResidueDiagram
  40. HookLengthDiagram
  41. RemoveRimHook
  42. AddRimHook
  43. Operations on partitions
  44. ECore
  45. IsECore
  46. EQuotient
  47. CombineEQuotientECore
  48. EWeight
  49. ERegularPartitions
  50. IsERegular
  51. ConjugatePartition
  52. ETopLadder
  53. Dominates
  54. LengthLexicographic
  55. Lexicographic
  56. ReverseDominance
  57. Miscellaneous functions on modules
  58. Specialized
  59. ERegulars
  60. SplitECores
  61. Coefficient for Sums of Modules
  62. InnerProduct
  63. SpechtPrettyPrint
  64. Semi--standard and standard tableaux
  65. SemiStandardTableaux
  66. StandardTableaux
  67. ConjugateTableau

71.1 Specht

Specht(e)
Specht(e, p)
Specht(e, p, val [,HeckeRing])

Let R be a field of characteristic 0, q a non--zero element of R, and let e be the smallest positive integer such that 1+q+ldots+q^e-1=0 (we set <e>=0 if no such integer exists). The record returned by Specht(e) allows calculations in the Grothendieck rings of the Hecke algebras H of type A which are defined over R and have parameter q. (The Hecke algebra is described in Chapter Iwahori-Hecke algebras; see also Hecke Hecke.) Below we also describe how to consider Hecke algebras defined over fields of positive characteristic.

Specht returns a record which contains, among other things, functions S , P, and D which correspond to the Specht modules, projective indecomposable modules, and the simple modules for the family of Hecke algebras determined by R and q. Specht allows manipulation of arbitrary linear combinations of these ``modules'', as well as a way of inducing and restricting them, ``multiplying'' them, and converting between these three natural bases of the Grothendieck ring. Multiplication of modules corresponds to taking a tensor product, and then inducing (thus giving a module for a larger Hecke algebra).

gap> RequirePackage("specht"); H:=Specht(5);
Specht(e=5, S(), P(), D(), Pq())
gap> H.D(3,2,1);
D(3,2,1)
gap> H.S( last );
S(6)-S(4,2)+S(3,2,1)
gap> InducedModule(H.P(3,2,1));
P(4,2,1)+P(3,3,1)+P(3,2,2)+2*P(3,2,1,1)
gap> H.S(last);
S(4,2,1)+S(3,3,1)+S(3,2,2)+2*S(3,2,1,1)+S(2,2,2,1)+S(2,2,1,1,1)
gap> H.D(3,1)*H.D(3);
D(7)+2*D(6,1)+D(5,2)+D(5,1,1)+2*D(4,3)+D(4,2,1)+D(3,3,1)
gap> RestrictedModule(last);
4*D(6)+3*D(5,1)+5*D(4,2)+2*D(4,1,1)+2*D(3,3)+2*D(3,2,1)
gap> H.S(last);
S(6)+3*S(5,1)+3*S(4,2)+2*S(4,1,1)+2*S(3,3)+2*S(3,2,1)
gap> H.P(last);
P(6)+3*P(5,1)+2*P(4,2)+2*P(4,1,1)+2*P(3,3) 

The way in which the partitions indexing the modules are printed can be changed using SpechtPrettyPrint SpechtPrettyPrint.

There is also a function Schur Schur for doing calculations with the q--Schur algebra. See DecompositionMatrix DecompositionMatrix, and CrystalDecompositionMatrix CrystalDecompositionMatrix.

This function requires the package ``specht'' (see RequirePackage). medskip

The functions H.S, H.P, and H.D

The functions H.S, H.P, and H.D return records which correspond to Specht modules, projective indecomposable modules, and simple modules respectively. Each of these three functions can be called in four different ways, as we now describe.

bigskip

H.S(mu) qquad H.P(mu) qquad H.D(mu)

In the first form, mu is a partition (either a list, or a sequence of integers), and the corresponding Specht module, PIM, or simple module (respectively), is returned.

gap> H.P(4,3,2);
P(4,3,2)

bigskip

H.S(x) qquad H.P(x) qquad H.D(x)

Here, x is an H--module. In this form, H.S rewrites x as a linear combination of Specht modules, if possible. Similarly, H.P and H.D rewrite x as a linear combination of PIMs and simple modules respectively. These conversions require knowledge of the relevant decomposition matrix of H; if this is not known then false is returned (over fields of characteristic zero, all of the decomposition matrices are known via the algorithm of [LLT]; various other decomposition matrices are included with Specht). For example, H.S(H.P(mu)) returns sum_nu d_numu S(nu), or false if some of these decomposition multiplicities are not known.

gap> H.D( H.P(4,3,2) );
D(5,3,1)+2*D(4,3,2)+D(2,2,2,2,1)
gap> H.S( H.D( H.S(1,1,1,1,1) ) );
-S(5)+S(4,1)-S(3,1,1)+S(2,1,1,1) 

As the last example shows, Specht does not always behave as expected. The reason for this is that Specht modules indexed by e--singular partitions can always be written as a linear combination of Specht modules which involve only e--regular partitions. As such, it is not always clear when two elements are equal in the Grothendieck ring. Consequently, to test whether two modules are equal you should first rewrite both modules in the D--basis; this is not done by Specht because it would be very inefficient.

bigskip

H.S(d, mu) qquad H.P(d, mu) qquad H.D(d, mu)

In the third form, d is a decomposition matrix and mu is a partition. This is useful when you are trying to calculate a new decomposition matrix d because it allows you to do calculations using the known entries of d to deduce information about the unknown ones. When used in this way, H.P and H.D use d to rewrite P(mu) and D(mu) respectively as a linear combination of Specht modules, and H.S uses d to write S(mu) as a linear combination of simple modules. If the values of the unknown entries in d are needed, false is returned.

gap> H:=Specht(3,3);   # e = 3, p = 3 = characteristic of 'R'
Specht(e=3, p=3, S(), P(), D())
gap>  d:=InducedDecompositionMatrix(DecompositionMatrix(H,14));;
# Inducing....
The following projectives are missing from <d>:
    [ 15 ]  [ 8, 7 ]
gap> H.P(d,4,3,3,2,2,1);
S(4,3,3,2,2,1)+S(4,3,3,2,1,1,1)+S(4,3,2,2,2,1,1)+S(3,3,3,2,2,1,1)
gap> H.S(d,7, 3, 3, 2);
D(11,2,1,1)+D(10,3,1,1)+D(8,5,1,1)+D(8,3,3,1)+D(7,6,1,1)+D(7,3,3,2)
gap> H.D(d,14,1);
false 

The final example returned false because the partitions (14,1) and (15) have the same 3--core (and P(15) is missing from d).

bigskip

H.S(d, x) qquad H.P(d, x) qquad H.D(d, x)

In the final form, d is a decomposition matrix and x is a module. All three functions rewrite x in their respective basis using d. Again this is only useful when you are trying to calculate a new decomposition matrix because, for any ``known'' decomposition matrix d, H.S(x) and H.S(d, x) are equivalent (and similarly for H.P and H.D).

gap> H.S(d, H.D(d,10,5) );
-S(13,2)+S(10,5)

71.2 Hecke algebras over fields of positive characteristic

The last example looked at Hecke algebras with parameter q=1 and R a field of characteristic~3 (so e=3); that is, the group algebra of the symmetric group over a field of characteristic 3. More, generally, the command Specht(p, p) can be used to consider the group algebras of the symmetric groups over fields of characteristic p (ie. e=p, and R a field of characteristic~p).

To consider Hecke algebras defined over arbitrary fields Specht must also be supplied with a valuation map val as an argument. The function val is a map from some PID into the natural numbers; at present it is needed only by functions which rely (at least implicitly), upon the q--analogue of Schaper's theorem. In general, val depends upon q and the characteristic of R; full details can be found in [JM2].

Over fields of characteristic zero, and in the symmetric group case, the function val is automatically defined by Specht. When R is a field of characteristic zero, val([i]_q) is 1 if e divides i and~0 otherwise (this is the valuation map associated to the prime ideal in C[v] generated by the e--th cyclotomic polynomial). When <q>=1 and R is a field of characteristic p, val is the usual p--adic valuation map.

As another example, if <q>=4 and R is a field of characteristic 5 (so <e>=2), then the valuation map sends the integer x to nu_5([4]_x) where [4]_x is interpreted as an integer and nu_5 is the usual 5--adic valuation. To consider this Hecke algebra one could proceed as follows:

gap> val:=function(x) local v;
>       x:=Sum([0..x-1],v->4^v);  # x-${>}$[x]\_q
>       v:=0; while x mod 5=0 do x:=x/5; v:=v+1; od;
>       return v;
>     end;;
gap> H:=Specht(2,5,val,"e2q4");
Specht(e=2, p=5, S(), P(), D(), HeckeRing="e2q4")

Notice the string ``e2q4'' which was also passed to Specht in this example. Although it is not strictly necessary, it is a good idea when using a ``non--standard'' valuation map val to specify the value of H.HeckeRing=HeckeRing. This string is used for internal bookkeeping by Specht; in particular, it is used to determine filenames when reading and saving decomposition matrices. If a ``standard'' valuation map is used then HeckeRing is set to the string ``e{<}e{>}p{<}p{>}''; otherwise it defaults to ``unknown''. The function SaveDecompositionMatrix will not save any decomposition matrix for any Hecke algebra H with H.HeckeRing=``unknown''.

71.3 The Fock space and Hecke algebras over fields of characteristic zero

For Hecke algebras H defined over fields of characteristic zero Lascoux, Leclerc and Thibon [LLT] have described an easy, inductive, algorithm for calculating the decomposition matrices of H. Their algorithm really calculates the canonical basis, or (global) crystal basis of the Fock space; results of Grojnowski--Lusztig [Gr] show that computing this basis is equivalent to computing the decomposition matrices of H (see also [A]).

The Fock space F is an (integrable) module for the quantum group U_q(widehat{sl}_{<e>}) of the affine special linear group. F is a free C[v]--module with basis the set of all Specht modules S(mu) for all partitions mu of all integers F = bigoplus_nge0bigoplus_muvdash nC[v] S(mu); here v=H.info.Indeterminate is an indeterminate over the integers (or strictly, C). The canonical basis elements Pq(mu) for the U_q(widehat{sl}_e)--submodule of F generated by the 0--partition are indexed by e--regular partitions mu. Moreover, under specialization, Pq(mu) maps to P(mu). An eloquent description of the algorithm for computing H.Pq(mu) can be found in [LLT].

To access the elements of the Fock space Specht provides the functions:

H.Pq(mu) qquad H.Sq(mu)

Notice that, unlike H.P and H.S the only arguments which H.Pq and H.Sq accept are partitions. (Given that our indeterminate is v these functions should really be called H.Pv and H.Sv; here ``q'' stands for ``quantum.)

The function H.Pq computes the canonical basis element Pq(mu) of the Fock space corresponding to the e--regular partition mu (there is a canonical basis for the whole of the Fock space [LT]; conjecturally, this basis can be used to compute the decomposition matrices for the q--Schur algebra over fields of characteristic zero). The second function returns a standard basis element S(mu) of F.

gap> H:=Specht(4);
Specht(e=4, S(), P(), D(), Pq())
gap> H.Pq(6,2); 
S(6,2)+v*S(5,3)
gap> RestrictedModule(last);
S(6,1)+(v + v^(-1))*S(5,2)+v*S(4,3)
gap> H.P(last);
P(6,1)+(v + v^(-1))*P(5,2)
gap> Specialized(last);
P(6,1)+2*P(5,2)
gap> H.Sq(5,3,2);
S(5,3,2)
gap> InducedModule(last,0);
v^(-1)*S(5,3,3)

The modules returned by H.Pq and H.Sq behave very much like elements of the Grothendieck ring of H; however, they should be considered as elements of the Fock space. The key difference is that when induced or restricted ``quantum'' analogues of induction and restriction are used. These analogues correspond to the action of U_q(widehat{sl}_{<e>}) on F [LLT].

In effect, the functions H.Pq and H.Sq allow computations in the Fock space, using the functions InducedModule InducedModule and RestrictedModule RestrictedModule. The functions H.S, H.P, and H.D can also be applied to elements of the Fock space, in which case they have the expected effect. In addition, any element of the Fock space can be specialized to give the corresponding element of the Grothendieck ring of H (it is because of this correspondence that we do not make a distinction between elements of the Fock space and the Grothendieck ring of H).

When working over fields of characteristic zero Specht will automatically calculate any canonical basis elements that it needs for computations in the Grothendieck ring of H. If you are not interested in the canonical basis elements you need never work with them directly. If, for some reason, you do not want Specht to use the canonical basis elements to calculate decomposition numbers then all you need to do is Unbind(H.Pq).

71.4 Schur

Schur(e)
Schur(e, p)
Schur(e, p, val [,HeckeRing])

This function behaves almost identically to the function Specht (see Specht), the only difference being that the three functions in the record S returned by Schur are called S.W, S.P, and S.F and that they correspond to the q-Weyl modules, the projective decomposable modules, and the simple modules of the q--Schur algebra respectively. Note that our labeling of these modules is non--standard, following that used by James in [J]. The standard labeling can be obtained from ours by replacing all partitions by their conjugates.

Almost all of the functions in Specht which accept a Specht record H will also accept a record S returned by Schur

In the current version of Specht the decomposition matrices of q--Schur algebras are not fully supported. The InducedDecompositionMatrix function can be applied to these matrices; however there are no additional routines available for calculating the columns corresponding to e--singular partitions. The decomposition matrices for the q--Schur algebras defined over a field of characteristic 0 for <n> le 10 are in the Specht libraries.

gap> S:=Schur(2);
Schur(e=2, W(), P(), F(), Pq())
gap> InducedDecompositionMatrix(DecompositionMatrix(S,3));
# The following projectives are missing from <d>:
#  [ 2, 2 ]
4     
|
 1                   # 'DecompositionMatrix'(S,4) returns the
3,1   
|
 1 1                 # full decomposition matrix. The point
2^2   
|
 . 1 .               # of this example is to emphasize the
2,1^2 
|
 1 1 . 1             # limitations of 'Schur'.
1^4   
|
 1 . . 1 1 

Note that when S is defined over a field of characteristic zero then it contains a function S.Pq for calculating canonical basis elements (see Specht Specht); currently S.Pq(mu) is implemented only for e--regular partitions. There is also a function H.Wq.

See also Specht Specht. This function requires the package ``specht'' (see RequirePackage).

71.5 DecompositionMatrix

DecompositionMatrix(H, n [,Ordering])
DecompositionMatrix(H, filename [,Ordering])

The function DecompositionMatrix returns the decomposition matrix D of 'H'(Sym_n) where H is a Hecke algebra record returned by the function Specht (or Schur). DecompositionMatrix first checks to see whether the required decomposition matrix exists as a library file (checking first in the current directory, next in the directory specified by SpechtDirectory, and finally in the Specht libraries). If H.Pq exists, DecompositionMatrix next looks for crystallized decomposition matrices (see CrystalDecompositionMatrix CrystalDecompositionMatrix). If the decomposition matrix d is not stored in te library DecompositionMatrix will calculate d when H is a Hecke algebra with a base field R of characteristic zero, and will return false otherwise (in which case the function CalculateDecompositionMatrix CalculateDecompositionMatrix can be used to force Specht to try and calculate this matrix).

For Hecke algebras defined over fields of characteristic zero, Specht uses the algorithm of [LLT] to calculate decomposition matrices (this feature can be disabled by unbinding H.Pq). The decomposition matrices for the q--Schur algebras for <n> le 10 are contained in the Specht library, as are those for the symmetric group over fields of positive characteristic when <n><15.

Once a decomposition matrix is known, Specht keeps an internal copy of it which is used by the functions H.S, H.P, and H.D; these functions also read decomposition matrix files as needed.

If you set the variable SpechtDirectory, then Specht will also search for decomposition matrix files in this directory. The files in the current directory override those in SpechtDirectory and those in the Specht libraries.

In the second form of the function, when a filename is supplied, DecompositionMatrix will read the decomposition matrix in the file filename, and this matrix will become Specht's internal copy of this matrix.

By default, the rows and columns of the decomposition matrices are ordered lexicographically. This can be changed by supplying DecompositionMatrix with an ordering function such as LengthLexicographic or ReverseDominance. You do not need to specify the ordering you want every time you call DecompositionMatrix; Specht will keep the same ordering until you change it again. This ordering can also be set ``by hand'' using the variable H.Ordering.

gap> DecompositionMatrix(Specht(3),6,LengthLexicographic);
6      
|
 1             
5,1    
|
 1 1           
4,2    
|
 . . 1         
3^2    
|
 . 1 . 1       
4,1^2  
|
 . 1 . . 1     
3,2,1  
|
 1 1 . 1 1 1   
2^3    
|
 1 . . . . 1   
3,1^3  
|
 . . . . 1 1   
2^2,1^2
|
 . . . . . . 1
2,1^4  
|
 . . . 1 . 1 . 
1^6    
| . . . 1 . . .

Once you have a decomposition matrix it is often nice to be able to print it. The on screen version is often good enough; there is also a TeX command which generates a LaTeX version. There are also functions for converting Specht decomposition matrices into GAP matrices and visa versa (see MatrixDecompositionMatrix MatrixDecompositionMatrix and DecompositionMatrixMatrix DecompositionMatrixMatrix).

Using the function InducedDecompositionMatrix (see InducedDecompositionMatrix), it is possible to induce a decomposition matrix. See also SaveDecompositionMatrix SaveDecompositionMatrix and IsNewIndecomposable IsNewIndecomposable, Specht Specht, Schur Schur, and CrystalDecompositionMatrix CrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.6 CrystalDecompositionMatrix

CrystalDecompositionMatrix(H, n [,Ordering])
CrystalDecompositionMatrix(H, filename [,Ordering])

This function is similar to DecompositionMatrix, except that it returns a crystallized decomposition matrix. The columns of decomposition matrices correspond to projective indecomposables; the columns of crystallized decomposition matrices correspond to the canonical basis elements of the Fock space (see Specht). Consequently, the entries in these matrices are polynomials (in v), and by specializing (ie. setting v equal to 1; see Specialized), the decomposition matrices of H are obtained (see Specht).

Crystallized decomposition matrices are defined only for Hecke algebras over a base field of characteristic zero. Unlike ``normal'' decomposition matrices, crystallized decomposition matrices cannot be induced.

gap> CrystalDecompositionMatrix(Specht(3), 6);
6      
|
   1                         
5,1    
|
   v   1                     
4,2    
|
   .   .   1                 
4,1^2  
|
   .   v   .   1             
3^2    
|
   .   v   .   .   1         
3,2,1  
|
   v v^2   .   v   v   1     
3,1^3  
|
   .   .   . v^2   .   v     
2^3    
|
 v^2   .   .   .   .   v     
2^2,1^2
|
   .   .   .   .   .   .   1
2,1^4  
|
   .   .   .   .   v v^2   . 
1^6    
|
   .   .   .   . v^2   .   . 
gap> Specialized(last);   # set 'v' equal to $1$.
6      
|
 1             
5,1    
|
 1 1           
4,2    
|
 . . 1         
4,1^2  
|
 . 1 . 1       
3^2    
|
 . 1 . . 1     
3,2,1  
|
 1 1 . 1 1 1   
3,1^3  
|
 . . . 1 . 1   
2^3    
|
 1 . . . . 1   
2^2,1^2
|
 . . . . . . 1
2,1^4  
|
 . . . . 1 1 . 
1^6    
| . . . . 1 . .

See also Specht Specht, Schur Schur, DecompositionMatrix DecompositionMatrix, and Specialized Specialized. This function requires the package ``specht'' (see RequirePackage).

71.7 DecompositionNumber

DecompositionNumber(H, mu, nu)
DecompositionNumber(d, mu, nu)

This function attempts to calculate the decomposition multiplicity of D(nu) in S(mu) (equivalently, the multiplicity of S(mu) in P(nu)). If P(nu) is known, we just look up the answer; if not DecompositionNumber tries to calculate the answer using ``row and column removal'' (see [J,Theorem~6.18]).

gap> H:=Specht(6);;
gap> DecompositionNumber(H,[6,4,2],[6,6]);  
0

This function requires the package ``specht'' (see RequirePackage).

71.8 Partitions in Specht

Many of the functions in Specht take partitions as arguments. Partitions are usually represented by lists in GAP. In Specht, all the functions which expect a partition will accept their argument either as a list or simply as a sequence of numbers. So, for example:

gap> H:=Specht(4);; H.S(H.P(6,4));
S(6,4)+S(6,3,1)+S(5,3,1,1)+S(3,3,2,1,1)+S(2,2,2,2,2)
gap> H.S(H.P([6,4]));
S(6,4)+S(6,3,1)+S(5,3,1,1)+S(3,3,2,1,1)+S(2,2,2,2,2) 

Some functions require more than one argument, but the convention still applies.

gap> ECore(3, [6,4,2]);
[ 6, 4, 2 ]
gap> ECore(3, 6,4,2);   
[ 6, 4, 2 ] 
gap> GoodNodes(3, 6,4,2);    
[ false, false, 3 ]
gap> GoodNodes(3, [6,4,2], 2);
3 

Basically, it never hurts to put the extra brackets in, and they can be omitted so long as this is not ambiguous. One function where the brackets are needed is DecompositionNumber; this is clear because the function takes two partitions as its arguments.

71.9 Inducing and restricting modules

Specht provides four functions InducedModule, RestrictedModule, SInducedModule and SRestrictedModule for inducing and restricting modules. All functions can be applied to Specht modules, PIMs, and simple modules. These functions all work by first rewriting all modules as a linear combination of Specht modules (or q--Weyl modules), and then inducing and restricting. Whenever possible the induced or restricted module will be written in the original basis.

All of these functions can also be applied to elements of the Fock space (see Specht); in which case they correspond to the action of the generators E_i and F_i of U_q(widehat{sl_e}) on F. There is also a function InducedDecompositionMatrix InducedDecompositionMatrix for inducing decomposition matrices.

71.10 InducedModule

InducedModule(x)
InducedModule(x, r_1 [,r_2, ...])

There is an natural embedding of 'H'(Sym_n) in 'H'(Sym_{n+1}) which in the usual way lets us define an induced 'H'(Sym_{n+1})--module for every 'H'(Sym_n)--module. The function InducedModule returns the induced modules of the Specht modules, principal indecomposable modules, and simple modules (more accurately, their image in the Grothendieck ring).

There is also a function SInducedModule (see SInducedModule) which provides a much faster way of r--inducing s times (and inducing s times).

Let mu be a partition. Then the induced module InducedModule(S(mu)) is easy to describe: it has the same composition factors as sum 'S'(<nu>) where nu runs over all partitions whose diagrams can be obtained by adding a single node to the diagram of mu.

gap> H:=Specht(2,2);
Specht(e=2, p=2, S(), P(), D())
gap> InducedModule(H.S(7,4,3,1));
S(8,4,3,1)+S(7,5,3,1)+S(7,4,4,1)+S(7,4,3,2)+S(7,4,3,1,1)
gap> InducedModule(H.P(5,3,1));
P(6,3,1)+2*P(5,4,1)+P(5,3,2)
gap> InducedModule(H.D(11,2,1));
# D(<x>), unable to rewrite <x> as a sum of simples
S(12,2,1)+S(11,3,1)+S(11,2,2)+S(11,2,1,1)

When inducing indecomposable modules and simple modules, InducedModule first rewrites these modules as a linear combination of Specht modules (using known decomposition matrices), and then induces this linear combination of Specht modules. If possible Specht then rewrites the induced module back in the original basis. Note that in the last example above, the decomposition matrix for Sym_{15} is not known by Specht; this is why InducedModule was unable to rewrite this module in the D--basis.

medskip

r--Induction

InducedModule(x, r_1 [, r_2, ...])

Two Specht modules S(mu) and S(nu) belong to the same block if and only if the corresponding partitions mu and nu have the same e--core [JM2] (see ECore). Because the e--core of a partition is determined by its (multiset of) e--residues, if S(mu) and S(nu) appear in InducedModule(S(tau)), for some partition tau, then S(mu) and S(nu) belong to the same block if and only if mu and nu can be obtained by adding a node of the same e--residue to the diagram of tau. The second form of InducedModule allows one to induce ``within blocks'' by only adding nodes of some fixed e--residue r; this is known as r-induction. Note that 0 le r.

gap> H:=Specht(4); InducedModule(H.S(5,2,1));
S(6,2,1)+S(5,3,1)+S(5,2,2)+S(5,2,1,1)
gap> InducedModule(H.S(5,2,1),0);
0*S()
gap> InducedModule(H.S(5,2,1),1);
S(6,2,1)+S(5,3,1)+S(5,2,1,1)
gap> InducedModule(H.S(5,2,1),2);
0*S()
gap> InducedModule(H.S(5,2,1),3);
S(5,2,2)

The function EResidueDiagram (EResidueDiagram), prints the diagram of mu, labeling each node with its e--residue. A quick check of this diagram confirms the answers above.

gap> EResidueDiagram(H,5,2,1);
  0  1  2  3  0
  3  0
  2

medskip

``Quantized'' induction

When InducedModule is applied to the canonical basis elements H.Pq(mu) (or more generally elements of the Fock space; see Specht), a ``quantum analogue'' of induction is applied. More precisely, the function InducedModule(*,i) corresponds to the action of the generator F_i of the quantum group U_q(widehat{sl_e}) on F [LLT].

gap> H:=Specht(3);; InducedModule(H.Pq(4,2),1,2);
S(6,2)+v*S(4,4)+v^2*S(4,2,2)
gap> H.P(last);
P(6,2)

See also SInducedModule SInducedModule, RestrictedModule RestrictedModule, and SRestrictedModule SRestrictedModule. This function requires the package ``specht'' (see RequirePackage).

71.11 SInducedModule

SInducedModule(x, s)
SInducedModule(x, s, r)

The function SInducedModule, standing for ``string induction'', provides a more efficient way of r--inducing s times (and a way of inducing s times if the residue r is omitted); r--induction is explained in InducedModule.

gap> H:=Specht(4);; SInducedModule(H.P(5,2,1),3);
P(8,2,1)+3*P(7,3,1)+2*P(7,2,2)+6*P(6,3,2)+6*P(6,3,1,1)+3*P(6,2,1,1,1)
+2*P(5,3,3)+P(5,2,2,1,1)
gap> SInducedModule(H.P(5,2,1),3,1);
P(6,3,1,1)
gap> InducedModule(H.P(5,2,1),1,1,1);
6*P(6,3,1,1)

Note that the multiplicity of each summand of InducedModule(x,r,...,r) is divisible by <s>! and that SInducedModule divides by this constant.

As with InducedModule this function can also be applied to elements of the Fock space (see Specht), in which case the quantum analogue of induction is used.

See also InducedModule InducedModule. This function requires the package ``specht'' (see RequirePackage).

71.12 RestrictedModule

RestrictedModule(x)
RestrictedModule(x, r_1 [, r_2, ...])

Given a module x for 'H'(Sym_n) RestrictedModule returns the corresponding module for 'H'(Sym_{n-1}). The restriction of the Specht module S(mu) is the linear combination of Specht modules sum 'S'(<nu>) where nu runs over the partitions whose diagrams are obtained by deleting a node from the diagram of mu. If only nodes of residue r are deleted then this corresponds to first restricting S(mu) and then taking one of the block components of the restriction; this process is known as r-restriction (cf. r--induction in InducedModule).

There is also a function SRestrictedModule (see SRestrictedModule) which provides a faster way of r--restricting s times (and restricting s times).

When more than one residue if given to RestrictedModule it returns RestrictedModule(x,r_1,r_2,...,r_k)= RestrictedModule(RestrictedModule(x,r_1),r_2,...,r_k) (cf. InducedModule InducedModule).

gap> H:=Specht(6);; RestrictedModule(H.P(5,3,2,1),4);
2*P(4,3,2,1)
gap> RestrictedModule(H.D(5,3,2),1); 
D(5,2,2) 

``Quantized'' restriction

As with InducedModule, when RestrictedModule is applied to the canonical basis elements H.Pq(mu) a quantum analogue of restriction is applied; this time, RestrictedModule(*,i) corresponds to the action of the generator E_i of U_q(widehat{sl_e}) on F [LLT].

See also InducedModule InducedModule, SInducedModule SInducedModule, and SRestrictedModule SRestrictedModule. This function requires the package ``specht'' (see RequirePackage).

71.13 SRestrictedModule

SRestrictedModule(x, s)
SRestrictedModule(x, s, r)

As with SInducedModule this function provides a more efficient way of r--restricting s times, or restricting s times if the residue r is omitted (cf. SInducedModule SInducedModule).

gap> H:=Specht(6);; SRestrictedModule(H.S(4,3,2),3);
3*S(4,2)+2*S(4,1,1)+3*S(3,3)+6*S(3,2,1)+2*S(2,2,2)
gap> SRestrictedModule(H.P(5,4,1),2,4);
P(4,4) 

See also InducedModule InducedModule, SInducedModule SInducedModule, and RestrictedModule RestrictedModule. This function requires the package ``specht'' (see RequirePackage).

71.14 Operations on decomposition matrices

Specht is a package for computing decomposition matrices; this section describes the functions available for accessing these matrices directly. In addition to decomposition matrices, Specht also calculates the ``crystallized decomposition matrices'' of [LLT], and the ``adjustment matrices'' introduced by James [J] (and Geck [G]).

Throughout Specht we place an emphasis on calculating the projective indecomposable modules, and hence upon the columns of decomposition matrices. This approach seems more efficient than the traditional approach of calculating decomposition matrices by rows; ideally both approaches should be combined (as is done by IsNewIndecomposable).

In principle, all decomposition matrices for all Hecke algebras defined over a field of characteristic zero are available from within Specht. In addition, the decomposition matrices for all q--Schur algebras with n le 10 and all values of e and the p--modular decomposition matrices of the symmetric groups Sym_n for n<15 are in the Specht library files.

If you are using Specht regularly to do calculations involving certain values of e it would be advantageous to have Specht calculate and save the first 20 odd decomposition matrices that you are interested in. So, for <e>=4 use the commands:

gap> H:=Specht(4);; for n in [8..20] do
>      SaveDecompositionMatrix(DecompositionMatrix(H,n));
>    od;

Alternatively, you could save the crystallized decomposition matrices. Note that for <n><2<e> the decomposition matrices are known (by Specht) and easy to compute.

71.15 InducedDecompositionMatrix

InducedDecompositionMatrix(d)

If d is the decomposition matrix of 'H'(Sym_n), then InducedDecompositionMatrix(d) attempts to calculate the decomposition matrix of 'H'(Sym_{n+1}). It does this by extracting each projective indecomposable from d and inducing these modules to obtain projective modules for 'H'(Sym_{n+1}). InducedDecompositionMatrix then tries to decompose these projectives using the function IsNewIndecomposable (see IsNewIndecomposable). In general there will be columns of the decomposition matrix which InducedDecompositionMatrix is unable to decompose and these will have to be calculated ``by hand''. InducedDecompositionMatrix prints a list of those columns of the decomposition matrix which it is unable to calculate (this list is also printed by the function MissingIndecomposables(d)).

gap> gap> d:=DecompositionMatrix(Specht(3,3),14);;
gap> InducedDecompositionMatrix(d);;
# Inducing....
The following projectives are missing from <d>:
    [ 15 ]  [ 8, 7 ]

Note that the missing indecomposables come in ``pairs'' which map to each other under the Mullineux map (see Mullineux Mullineux).

Almost all of the decomposition matrices included in Specht were calculated directly by InducedDecompositionMatrix. When n is ``small'' InducedDecompositionMatrix is usually able to return the full decomposition matrix for 'H'(Sym_{n+1}).

Finally, although the InducedDecompositionMatrix can also be applied to the decomposition matrices of the q--Schur algebras (see Schur Schur), InducedDecompositionMatrix is much less successful in inducing these decomposition matrices because it contains no special routines for dealing with the indecomposable modules of the q--Schur algebra which are indexed by e--singular partitions. Note also that we use a non--standard labeling of the decomposition matrices of q--Schur algebras; see Schur.

71.16 IsNewIndecomposable

IsNewIndecomposable(d, x [,mu])

IsNewIndecomposable is the function which does all of the hard work when the function InducedDecompositionMatrix is applied to decomposition matrices (see InducedDecompositionMatrix). Given a projective module x, IsNewIndecomposable returns true if it is able to show that x is indecomposable (and this indecomposable is not already listed in d), and false otherwise. IsNewIndecomposable will also print a brief description of its findings, giving an upper and lower bound on the first decomposition number mu for which it is unable to determine the multiplicity of S(mu) in x.

IsNewIndecomposable works by running through all of the partitions nu such that P(nu) could be a summand of x and it uses various results, such as the q-Schaper theorem of [JM2] (see Schaper Schaper), the Mullineux map (see Mullineux Mullineux), and inducing simple modules, to determine if P(nu) does indeed split off. In addition, if d is the decomposition matrix for 'H'(Sym_n) then IsNewIndecomposable will probably use some of the decomposition matrices of 'H'(Sym_m) for m le n, if they are known. Consequently it is a good idea to save decomposition matrices as they are calculated (see SaveDecompositionMatrix).

For example, in calculating the 2--modular decomposition matrices of Sym_{r} the first projective which InducedDecompositionMatrix is unable to calculate is P(10).

gap> H:=Specht(2,2);; 
gap> d:=InducedDecompositionMatrix(DecompositionMatrix(H,9));; 
# Inducing.
# The following projectives are missing from <d>:
#  [ 10 ]

(In fact, given the above commands, Specht will return the full decomposition matrix for Sym_{10} because this matrix is in the library; these were the commands that I used to calculate the decomposition matrix in the library.)

By inducing P(9) we can find a projective H--module which contains P(10). We can then use IsNewIndecomposable to try and decompose this induced module into a sum of PIMs.

gap> SpechtPrettyPrint(); x:=InducedModule(H.P(9),1);
S(10)+S(9,1)+S(8,2)+2S(8,1^2)+S(7,3)+2S(7,1^3)+3S(6,3,1)+3S(6,2^2)
+4S(6,2,1^2)+2S(6,1^4)+4S(5,3,2)+5S(5,3,1^2)+5S(5,2^2,1)+2S(5,1^5)
+2S(4^2,2)+2S(4^2,1^2)+2S(4,3^2)+5S(4,3,1^3)+2S(4,2^3)+5S(4,2^2,1^2)
+4S(4,2,1^4)+2S(4,1^6)+2S(3^3,1)+2S(3^2,2^2)+4S(3^2,2,1^2)
+3S(3^2,1^4)+3S(3,2^2,1^3)+2S(3,1^7)+S(2^3,1^4)+S(2^2,1^6)+S(2,1^8)
+S(1^10)
gap> IsNewIndecomposable(d,x);
# The multiplicity of S(6,3,1) in P(10) is at least 1 and at most 2.
false
gap> x;
S(10)+S(9,1)+S(8,2)+2S(8,1^2)+S(7,3)+2S(7,1^3)+2S(6,3,1)+2S(6,2^2)
+3S(6,2,1^2)+2S(6,1^4)+3S(5,3,2)+4S(5,3,1^2)+4S(5,2^2,1)+2S(5,1^5)
+2S(4^2,2)+2S(4^2,1^2)+2S(4,3^2)+4S(4,3,1^3)+2S(4,2^3)+4S(4,2^2,1^2)
+3S(4,2,1^4)+2S(4,1^6)+2S(3^3,1)+2S(3^2,2^2)+3S(3^2,2,1^2)
+2S(3^2,1^4)+2S(3,2^2,1^3)+2S(3,1^7)+S(2^3,1^4)+S(2^2,1^6)+S(2,1^8)
+S(1^10)

Notice that some of the coefficients of the Specht modules in x have changed; this is because IsNewIndecomposable was able to determine that the multiplicity of S(6,3,1) was at most 2 and so it subtracted one copy of P(6,3,1) from x.

In this case, the multiplicity of S(6,3,1) in P(10) is easy to resolve because general theory says that this multiplicity must be odd. Therefore, x-'P'(6,3,1) is projective. After subtracting P(6,3,1) from x we again use IsNewIndecomposable to see if x is now indecomposable. We can tell IsNewIndecomposable that all of the multiplicities up to and including S(6,3,1) have already been checked by giving it the addition argument mu=[6,3,1].

gap> x:=x-H.P(d,6,3,1);; IsNewIndecomposable(d,x,6,3,1);
true

Consequently, <x>='P'(10) and we add it to the decomposition matrix d (and save it).

gap> AddIndecomposable(d,x); SaveDecompositionMatrix(d);

A full description of what IsNewIndecomposable does can be found by reading the comments in specht.g. Any suggestions or improvements on this function would be especially welcome.

See also DecompositionMatrix DecompositionMatrix and InducedDecompositionMatrix InducedDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.17 InvertDecompositionMatrix

InvertDecompositionMatrix(d)

Returns the inverse of the (e--regular part of) d, where d is a decomposition matrix, or crystallized decomposition matrix, of a Hecke algebra or q--Schur algebra. If part of the decomposition matrix d is unknown then InvertDecompositionMatrix will invert as much of d as possible.

gap> H:=Specht(4);; d:=CrystalDecompositionMatrix(H,5);;
gap> InvertDecompositionMatrix(d);
5    
|
   1                     
4,1  
|
   .   1                 
3,2  
|
  -v   .   1             
3,1^2
|
   .   .   .   1         
2^2,1
|
 v^2   .  -v   .   1     
2,1^3
| . . . . . 1

See also DecompositionMatrix DecompositionMatrix, and CrystalDecompositionMatrix CrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.18 AdjustmentMatrix

AdjustmentMatrix(dp, d)

James [J] noticed, and Geck [G] proved, that the decomposition matrices dp for Hecke algebras defined over fields of positive characteristic admit a factorization dp = d * a where d is a decomposition matrix for a suitable Hecke algebra defined over a field of characteristic zero, and a is the so--called adjustment matrix. This function returns the adjustment matrix a.

gap> H:=Specht(2);; Hp:=Specht(2,2);;
gap> d:=DecompositionMatrix(H,13);; dp:=DecompositionMatrix(Hp,13);;
gap> a:=AdjustmentMatrix(dp,d);
13     
|
 1
12,1   
|
 . 1
11,2   
|
 1 . 1
10,3   
|
 . . . 1
10,2,1 
|
 . . . . 1
9,4    
|
 1 . 1 . . 1
9,3,1  
|
 2 . . . . . 1
8,5    
|
 . 1 . . . . . 1
8,4,1  
|
 1 . . . . . . . 1
8,3,2  
|
 . 2 . . . . . 1 . 1
7,6    
|
 1 . . . . 1 . . . . 1
7,5,1  
|
 . . . . . . 1 . . . . 1
7,4,2  
|
 1 . 1 . . 1 . . . . 1 . 1
7,3,2,1
|
 . . . . . . . . . . . . . 1
6,5,2  
|
 . 1 . . . . . 1 . 1 . . . . 1
6,4,3  
|
 2 . . . 1 . . . . . . . . . . 1
6,4,2,1
|
 . 2 . 1 . . . . . . . . . . . . 1
5,4,3,1
|
 4 . 2 . . . . . . . . . . . . . . 1 
gap> MatrixDecompositionMatrix(dp)=
>           MatrixDecompositionMatrix(d)*MatrixDecompositionMatrix(a);
true 

In the last line we have checked our calculation.

See also DecompositionMatrix DecompositionMatrix, and CrystalDecompositionMatrix CrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.19 SaveDecompositionMatrix

SaveDecompositionMatrix(d)
SaveDecompositionMatrix(d, filename)

The function SaveDecompositionMatrix saves the decomposition matrix d. After a decomposition matrix has been saved, the functions H.S, H.P, and H.D will automatically access it as needed. So, for example, before saving d in order to retrieve the indecomposable P(mu) from d it is necessary to type H.P(d, mu); once d has been saved, the command H.P(mu) suffices.

Since InducedDecompositionMatrix(d) consults the decomposition matrices for smaller n, if they are available, it is advantageous to save decomposition matrices as they are calculated. For example, over a field of characteristic~5, the decomposition matrices for the symmetric groups Sym_n with n le 20 can be calculated as follows:

gap> H:=Specht(5,5);;
gap> d:=DecompositionMatrix(H,9);; 
gap> for r in [10..20] do
>      d:=InducedDecompositionMatrix(d); 
>      SaveDecompositionMatrix(d); 
>    od; 

If your Hecke algebra record H is defined using a non--standard valuation map (see Specht) then it is also necessary to set the string ``H.HeckeRing'', or to supply the function with a filename before it will save your matrix. SaveDecompositionMatrix will also save adjustment matrices and the various other matrices that appear in Specht (they can be read back in using DecompositionMatrix). Each matrix has a default filename which you can over ride by supplying a filename. Using non--standard file names will stop Specht from automatically accessing these matrices in future.

See also DecompositionMatrix DecompositionMatrix DecompositionMatrix and CrystalDecompositionMatrix CrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.20 CalculateDecompositionMatrix

CalculateDecompositionMatrix(H,n)

CalculateDecompositionMatrix(H,n) is similar to the function DecompositionMatrix DecompositionMatrix in that both functions try to return the decomposition matrix d of 'H'(Sym_n); the difference is that this function tries to calculate this matrix whereas the later reads the matrix from the library files (in characteristic zero both functions apply the algorithm of [LLT] to compute~d). In effect this function is only needed when working with Hecke algebras defined over fields of positive characteristic (or when you wish to avoid the libraries).

For example, if you want to do calculations with the decomposition matrix of the symmetrix group Sym_{15} over a field of characteristic two, DecompositionMatrix returns false whereas CalculateDecompositionMatrix; returns a part of the decomposition matrix.

gap> H:=Specht(2,2);
Specht(e=2, p=2, S(), P(), D())
gap> d:=DecompositionMatrix(H,15); 
# This decomposition matrix is not known; use CalculateDecompositionMatrix()
# or InducedDecompositionMatrix() to calculate with this matrix.
false
gap> d:=CalculateDecompositionMatrix(H,15);;
# Projective indecomposable P(6,4,3,2) not known.
# Projective indecomposable P(6,5,3,1) not known.
...
gap> MissingIndecomposables(d);
The following projectives are missing from <d>:
    [ 15 ]  [ 14, 1 ]  [ 13, 2 ]  [ 12, 3 ]  [ 12, 2, 1 ]  [ 11, 4 ]  
[ 11, 3, 1 ]  [ 10, 5 ]  [ 10, 4, 1 ]  [ 10, 3, 2 ]  [ 9, 6 ]  [ 9, 5, 1 ]
[ 9, 4, 2 ]  [ 9, 3, 2, 1 ]  [ 8, 7 ]  [ 8, 6, 1 ]  [ 8, 5, 2 ]  [ 8, 4, 3]  
[ 8, 4, 2, 1 ]  [ 7, 6, 2 ]  [ 7, 5, 3 ]  [ 7, 5, 2, 1 ]  [ 7, 4, 3, 1 ]  
[ 6, 5, 4 ]  [ 6, 5, 3, 1 ]  [ 6, 4, 3, 2 ]

Actually, you are much better starting with the decompositon matrix of Sym_{14} and then applying InducedDecompositionMatrix to this matrix.

See also DecompositionMatrix DecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.21 MatrixDecompositionMatrix

MatrixDecompositionMatrix(d)

qqqqReturns the GAP matrix corresponding to the Specht decomposition matrix d. The rows and columns of d are ordered by H.Ordering.

gap> MatrixDecompositionMatrix(DecompositionMatrix(Specht(3),5));      
[ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 1, 1, 0, 0 ], 
  [ 0, 0, 0, 1, 0 ], [ 1, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1 ], 
  [ 0, 0, 1, 0, 0 ] ]

See also DecompositionMatrix DecompositionMatrix and DecompositionMatrixMatrix DecompositionMatrixMatrix. This function requires the package ``specht'' (see RequirePackage).

71.22 DecompositionMatrixMatrix

DecompositionMatrixMatrix(H, m, n)

Given a Hecke algebra H, a GAP matrix m, and an integer n this function returns the Specht decomposition matrix corresponding to m. If p is the number of partitions of n and r the number of e--regular partitions of n, then m must be either <r>times<r>, <p>times<r>, or <p>times<p>. The rows and columns of m are assumed to be indexed by partitions ordered by H.Ordering (see Specht).

gap> H:=Specht(3);;
gap> m:=[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 1, 0 ], 
>         [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ];;
gap> DecompositionMatrixMatrix(H,m,4);
4    
|
 1       
3,1  
|
 . 1     
2^2  
|
 1 . 1   
2,1^2
|
 . . . 1
1^4  
| . . 1 .

See also DecompositionMatrix DecompositionMatrix and MatrixDecompositionMatrix MatrixDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.23 AddIndecomposable

AddIndecomposable(d, x)

AddIndecomposable(d, x) inserts the indecomposable module x into the decomposition matrix d. If d already contains the indecomposable x then a warning is printed. The function AddIndecomposable also calculates Mullineux(x) (see Mullineux) and adds this indecomposable to d (or checks to see that it agrees with the corresponding entry of d if this indecomposable is already by d).

See IsNewIndecomposable IsNewIndecomposable for an example. See also DecompositionMatrix DecompositionMatrix and CrystalDecompositionMatrix CrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.24 RemoveIndecomposable

RemoveIndecomposable(d, mu)

The function RemoveIndecomposable removes the column from d which corresponds to P(mu). This is sometimes useful when trying to calculate a new decomposition matrix using Specht and want to test a possible candidate for a yet to be identified PIM.

See also DecompositionMatrix DecompositionMatrix and CrystalDecompositionMatrix CrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.25 MissingIndecomposables

MissingIndecomposables(d)

The function MissingIndecomposables prints the list of partitions corresponding to the indecomposable modules which are not listed in d.

See also DecompositionMatrix DecompositionMatrix and CrystalDecompositionMatrix CrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.26 Calculating dimensions

Specht has two functions for calculating the dimensions of modules of Hecke algebras; SimpleDimension and SpechtDimension. As yet, Specht does not know how to calculate the dimensions of modules for q--Schur algebras (these depend upon q).

71.27 SimpleDimension

SimpleDimension(d)
SimpleDimension(H, n)
SimpleDimension(H|d, mu)

In the first two forms, SimpleDimension prints the dimensions of all of the simple modules specified by d or for the Hecke algebra 'H'(Sym_n) respectively. If a partition mu is supplied, as in the last form, then the dimension of the simple module D(mu) is returned. At present the function is not implemented for the simple modules of the q--Schur algebras.

gap> H:=Specht(6);;
gap> SimpleDimension(H,11,3);
272
gap> d:=DecompositionMatrix(H,5);; SimpleDimension(d,3,2);
5
gap> SimpleDimension(d);
5     : 1
4,1   : 4
3,2   : 5
3,1^2 : 6
2^2,1 : 5
2,1^3 : 4
1^5   : 1

This function requires the package ``specht'' (see RequirePackage).

71.28 SpechtDimension

SpechtDimension(mu)

Calculates the dimension of the Specht module S(mu), which is equal to the number of standard mu-tableaux; the answer is given by the hook length formula (see [JK]).

gap> SpechtDimension(6,3,2,1);
5632

See also SimpleDimension SimpleDimension. This function requires the package ``specht'' (see RequirePackage).

71.29 Combinatorics on Young diagrams

These functions range from the representation theoretic q--Schaper theorem and Kleshchev's algorithm for the Mullineux map through to simple combinatorial operations like adding and removing rim hooks from Young diagrams.

71.30 Schaper

Schaper(H, mu)

Given a partition mu, and a Hecke algebra H, Schaper returns a linear combination of Specht modules which have the same composition factors as the sum of the modules in the ``Jantzen filtration'' of S(mu); see [JM2]. In particular, if nu strictly dominates mu then D(nu) is a composition factor of S(mu) if and only if it is a composition factor of Schaper(mu).

Schaper uses the valuation map H.valuation attached to H (see Specht and [JM2]).

One way in which the q--Schaper theorem can be applied is as follows. Suppose that we have a projective module x, written as a linear combination of Specht modules, and suppose that we are trying to decide whether the projective indecomposable P(mu) is a direct summand of x. Then, providing that we know that P(nu) is not a summand of x for all (e--regular) partitions nu which strictly dominate mu (see Dominates), P(mu) is a summand of x if and only if InnerProduct(Schaper(H,mu),x) is non--zero (note, in particular, that we don't need to know the indecomposable P(mu) in order to perform this calculation).

The q--Schaper theorem can also be used to check for irreduciblity; in fact, this is the basis for the criterion employed by IsSimpleModule.

gap> H:=Specht(2);;
gap> Schaper(H,9,5,3,2,1);
S(17,2,1)-S(15,2,1,1,1)+S(13,2,2,2,1)-S(11,3,3,2,1)+S(10,4,3,2,1)-S(9,8,3)
-S(9,8,1,1,1)+S(9,6,3,2)+S(9,6,3,1,1)+S(9,6,2,2,1)
gap> Schaper(H,9,6,5,2);  
0*S(0)

The last calculation shows that S(9,6,5,2) is irreducible when R is a field of characteristic 0 and e=2 (cf. IsSimpleModule(H,9,6,5,2)).

This function requires the package ``specht'' (see RequirePackage).

71.31 IsSimpleModule

IsSimpleModule(H, mu)

mu an e--regular partition.

Given an e--regular partition mu, IsSimpleModule(H, mu) returns true if S(mu) is simple and false otherwise. This calculation uses the valuation function H.valuation; see Specht. Note that the criterion used by IsSimpleModule is completely combinatorial; it is derived from the q--Schaper theorem [JM2].

gap> H:=Specht(3);;
gap> IsSimpleModule(H,45,31,24);
false

See also Schaper Schaper. This function requires the package ``specht'' (see RequirePackage).

71.32 Mullineux

Mullineux(e|H, mu)
Mullineux(d, mu)
Mullineux(x)

Given an integer e, or a Specht record H, and a partition mu, Mullineux(e, mu) returns the image of mu under the Mullineux map; which we now explain.

The sign representation D(1^n) of the Hecke algebra is the (one dimensional) representation sending T_w to (-1)^{ell(w)}. The Hecke algebra H is not a Hopf algebra so there is no well defined action of H upon the tensor product of two H--modules; however, there is an outer automorphism # of H which corresponds to tensoring with D(1^n). This sends an irreducible module 'D'(<mu>) to an irreducible 'D'(<mu>)^#cong 'D'(<mu^#>) for some e--regular partition mu^#. In the symmetric group case, Mullineux gave a conjectural algorithm for calculating mu^#; consequently the map sending mu to mu^# is known as the Mullineux map.

Deep results of Kleshchev [K] for the symmetric group give another (proven) algorithm for calculating the partition mu^# (Ford and Kleshchev have deduced Mullineux's conjecture from this). Using the canonical basis, it was shown by [LLT] that the natural generalization of Kleshchev's algorithm to H gives the Mullineux map for Hecke algebras over fields of characteristic zero. The general case follows from this, so the Mullineux map is now known for all Hecke algebras.

Kleshchev's map is easy to describe; he proved that if gns is any good node sequence for mu, then the sequence obtained from gns by replacing each residue r by -rbmod e is a good node sequence for mu^# (see GoodNodeSequence GoodNodeSequence).

gap> Mullineux(Specht(2),12,5,2); 
[ 12, 5, 2 ]
gap> Mullineux(Specht(4),12,5,2);
[ 4, 4, 4, 2, 2, 1, 1, 1 ]
gap> Mullineux(Specht(6),12,5,2);
[ 4, 3, 2, 2, 2, 2, 2, 1, 1 ]
gap> Mullineux(Specht(8),12,5,2);
[ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 ]
gap> Mullineux(Specht(10),12,5,2);
[ 3, 3, 3, 3, 2, 1, 1, 1, 1, 1 ]

Mullineux(d, mu)

The Mullineux map can also be calculated using a decomposition matrix. To see this recall that ``tensoring'' a Specht module S(mu) with the sign representation yields a module isomorphic to the dual of S(lambda), where lambda is the partition conjugate to mu. It follows that d_{munu}=d_{lambdanu^#} for all e--regular partitions nu. Therefore, if mu is the last partition in the lexicographic order such that d_{munu}ne0 then we must have nu^#=lambda. The second form of Mullineux uses d to calculate mu^# rather than the Kleshchev--[LLT] result.

Mullineux(x)

In the third form, x is a module, and Mullineux returns <x>^#, the image of x under #. Note that the above remarks show that P(mu) is mapped to P(mu^#) via the Mullineux map; this observation is useful when calculating decomposition matrices (and is used by the function InducedDecompositionMatrix).

See also GoodNodes GoodNodes and GoodNodeSequence GoodNodeSequence . This function requires the package ``specht'' (see RequirePackage).

71.33 GoodNodes

GoodNodes(e|H, mu)
GoodNodes(e|H, mu, r)

Given a partition and an integer e, Kleshchev [K] defined the notion of good node for each residue r (0 le r). When e is prime and mu is e--regular, Kleshchev showed that the good nodes describe the restriction of the socle of D(mu) in the symmetric group case. Brundan~[B] has recently generalized this result to the Hecke algebra.

By definition, there is at most one good node for each residue r, and this node is a removable node (in the diagram of mu). The function GoodNodes returns a list of the rows of mu which end in a good node; the good node of residue r (if it exists) is the (r+1)--st element in this list. In the second form, the number of the row which ends with the good node of residue r is returned; or false if there is no good node of residue r.

gap> GoodNodes(5,[5,4,3,2]);
[ false, false, 2, false, 1 ]
gap> GoodNodes(5,[5,4,3,2],0);
false
gap> GoodNodes(5,[5,4,3,2],4);
1

The good nodes also determine the Kleshchev--Mullineux map (see GoodNodeSequence GoodNodeSequence and Mullineux Mullineux). This function requires the package ``specht'' (see RequirePackage).

71.34 GoodNodeSequence

GoodNodeSequence(e|H, mu)
GoodNodeSequences(e|H, mu)

mu an e--regular partition.

Given an e--regular partition mu of n, a good node sequence for mu is a sequence gns of n residues such that mu has a good node of residue r, where r is the last residue in gns, and the first n-1 residues in gns are a good node sequence for the partition obtained from mu by deleting its (unique) good node with residue r (see GoodNodes GoodNodes). In general, mu will have more than one good node sequence; however, any good node sequence uniquely determines mu (see PartitionGoodNodeSequence PartitionGoodNodeSequence).

gap> H:=Specht(4);; GoodNodeSequence(H,4,3,1);
[ 0, 3, 1, 0, 2, 2, 1, 3 ]
gap> GoodNodeSequence(H,4,3,2);
[ 0, 3, 1, 0, 2, 2, 1, 3, 3 ]
gap> GoodNodeSequence(H,4,4,2);  
[ 0, 3, 1, 0, 2, 2, 1, 3, 3, 2 ]
gap> GoodNodeSequence(H,5,4,2);
[ 0, 3, 1, 0, 2, 2, 1, 3, 3, 2, 0 ]

The function GoodNodeSequences returns the list of all good node sequences for mu.

gap> GoodNodeSequences(H,5,2,1);
[ [ 0, 1, 2, 3, 3, 2, 0, 0 ], [ 0, 3, 1, 2, 2, 3, 0, 0 ], 
  [ 0, 1, 3, 2, 2, 3, 0, 0 ], [ 0, 1, 2, 3, 3, 0, 2, 0 ], 
  [ 0, 1, 2, 3, 0, 3, 2, 0 ], [ 0, 1, 2, 3, 3, 0, 0, 2 ], 
  [ 0, 1, 2, 3, 0, 3, 0, 2 ] ] 

The good node sequences determine the Mullineux map (see GoodNodes GoodNodes and Mullineux Mullineux). This function requires the package ``specht'' (see RequirePackage).

71.35 PartitionGoodNodeSequence

PartitionGoodNodeSequence(e|H, gns)

Given a good node sequence gns (see GoodNodeSequence GoodNodeSequence), this function returns the unique e--regular partition corresponding to gns (or false if in fact gns is not a good node sequence).

gap> H:=Specht(4);; 
gap> PartitionGoodNodeSequence(H,0, 3, 1, 0, 2, 2, 1, 3, 3, 2);
[ 4, 4, 2 ]

See also GoodNodes GoodNodes, GoodNodeSequence GoodNodeSequence and Mullineux Mullineux. This function requires the package ``specht'' (see RequirePackage).

71.36 GoodNodeLatticePath

GoodNodeLatticePath(e|H, mu)
GoodNodeLatticePaths(e|H, mu)
LatticePathGoodNodeSequence(e|H, gns)

The function GoodNodeLatticePath returns a sequence of partitions which give a path in the e--good partition lattice from the empty partition to mu. The second function returns the list of all paths in the e--good partition lattice which end in mu, and the third function returns the path corresponding to a given good node sequence gns.

gap> GoodNodeLatticePath(3,3,2,1);
[ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 1, 1 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ]
gap> GoodNodeLatticePaths(3,3,2,1);
[ [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 1, 1 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ], 
  [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 2 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ] ]
gap> GoodNodeSequence(4,6,3,2);
[ 0, 3, 1, 0, 2, 2, 3, 3, 0, 1, 1 ]
gap> LatticePathGoodNodeSequence(4,last);
[ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 2 ], [ 3, 2 ], [ 3, 2, 1 ], [ 4, 2, 1 ], 
  [ 4, 2, 2 ], [ 5, 2, 2 ], [ 6, 2, 2 ], [ 6, 3, 2 ] ] 

See also GoodNodes GoodNodes. This function requires the package ``specht'' (see RequirePackage).

71.37 LittlewoodRichardsonRule

LittlewoodRichardsonRule(mu, nu)
LittlewoodRichardsonCoefficient(mu, nu, tau)

Given partitions mu of n and nu of m the module 'S'(<mu>) otimes 'S'(<nu>) is naturally an 'H'(Sym_ntimesSym_m)-module and, by inducing, we obtain an 'H'(Sym_{n+m})-module. This module has the same composition factors as sum_nu a_munu^lambda S(lambda), where the sum runs over all partitions lambda of n+m and the integers a_{munu}^lambda are the Littlewood--Richardson coefficients. The integers a_{munu}^lambda can be calculated using a straightforward combinatorial algorithm known as the Littlewood--Richardson rule (see [JK]).

The function LittlewoodRichardsonRule returns an (unordered) list of partitions of n+m in which each partition lambda occurs a_{munu}^lambda times. The Littlewood-Richardson coefficients are independent of e; they can be read more easily from the computation S(mu)*S(nu).

gap> H:=Specht(0);;    # the generic Hecke algebra with 'R'=*C*['q']
gap> LittlewoodRichardsonRule([3,2,1],[4,2]);
[ [ 4, 3, 2, 2, 1 ],[ 4, 3, 3, 1, 1 ],[ 4, 3, 3, 2 ],[ 4, 4, 2, 1, 1 ],
 [ 4, 4, 2, 2 ],[ 4, 4, 3, 1 ],[ 5, 2, 2, 2, 1 ],[ 5, 3, 2, 1, 1 ], 
  [ 5, 3, 2, 2 ],[ 5, 4, 2, 1 ],[ 5, 3, 2, 1, 1 ],[ 5, 3, 3, 1 ], 
  [ 5, 4, 1, 1, 1 ],[ 5, 4, 2, 1 ],[ 5, 5, 1, 1 ],[ 5, 3, 2, 2 ], 
  [ 5, 3, 3, 1 ],[ 5, 4, 2, 1 ],[ 5, 4, 3 ],[ 5, 5, 2 ],[ 6, 2, 2, 1, 1],
  [ 6, 3, 1, 1, 1 ],[ 6, 3, 2, 1 ],[ 6, 4, 1, 1 ],[ 6, 2, 2, 2 ], 
  [ 6, 3, 2, 1 ],[ 6, 4, 2 ],[ 6, 3, 2, 1 ],[ 6, 3, 3 ],[ 6, 4, 1, 1 ],
  [ 6, 4, 2 ], [ 6, 5, 1 ], [ 7, 2, 2, 1 ], [ 7, 3, 1, 1 ], [ 7, 3, 2 ], 
  [ 7, 4, 1 ] ]
gap> H.S(3,2,1)*H.S(4,2);
S(7,4,1)+S(7,3,2)+S(7,3,1,1)+S(7,2,2,1)+S(6,5,1)+2*S(6,4,2)+2*S(6,4,1,1)
+S(6,3,3)+3*S(6,3,2,1)+S(6,3,1,1,1)+S(6,2,2,2)+S(6,2,2,1,1)+S(5,5,2)
+S(5,5,1,1)+S(5,4,3)+3*S(5,4,2,1)+S(5,4,1,1,1)+2*S(5,3,3,1)+2*S(5,3,2,2)
+2*S(5,3,2,1,1)+S(5,2,2,2,1)+S(4,4,3,1)+S(4,4,2,2)+S(4,4,2,1,1)+S(4,3,3,2)
+S(4,3,3,1,1)+S(4,3,2,2,1)
gap> LittlewoodRichardsonCoefficient([3,2,1],[4,2],[5,4,2,1]);
3

The function LittlewoodRichardsonCoefficient returns a single Littlewood--Richardson coefficient (although you are really better off asking for all of them, since they will all be calculated anyway).

See also InducedModule InducedModule and InverseLittlewoodRichardsonRule InverseLittlewoodRichardsonRule. This function requires the package ``specht'' (see RequirePackage).

71.38 InverseLittlewoodRichardsonRule

InverseLittlewoodRichardsonRule(tau)

Returns a list of all pairs of partitions [mu,nu] such that the Littlewood-Richardson coefficient a_{munu}^tau is non-zero (see LittlewoodRichardsonRule). The list returned is unordered and [mu,nu] will appear a_{munu}^tau times in it.

gap> InverseLittlewoodRichardsonRule([3,2,1]);
[ [ [  ],[ 3, 2, 1 ] ],[ [ 1 ],[ 3, 2 ] ],[ [ 1 ],[ 2, 2, 1 ] ], 
  [ [ 1 ],[ 3, 1, 1 ] ],[ [ 1, 1 ],[ 2, 2 ] ],[ [ 1, 1 ],[ 3, 1 ] ], 
  [ [ 1, 1 ],[ 2, 1, 1 ] ],[ [ 1, 1, 1 ],[ 2, 1 ] ],[ [ 2 ],[ 2, 2 ] ], 
  [ [ 2 ],[ 3, 1 ] ],[ [ 2 ],[ 2, 1, 1 ] ],[ [ 2, 1 ],[ 3 ] ], 
  [ [ 2, 1 ],[ 2, 1 ] ],[ [ 2, 1 ],[ 2, 1 ] ],[ [ 2, 1 ],[ 1, 1, 1 ] ], 
  [ [ 2, 1, 1 ],[ 2 ] ],[ [ 2, 1, 1 ],[ 1, 1 ] ],[ [ 2, 2 ],[ 2 ] ], 
  [ [ 2, 2 ],[ 1, 1 ] ],[ [ 2, 2, 1 ],[ 1 ] ],[ [ 3 ],[ 2, 1 ] ], 
  [ [ 3, 1 ],[ 2 ] ],[ [ 3, 1 ],[ 1, 1 ] ],[ [ 3, 1, 1 ],[ 1 ] ], 
  [ [ 3, 2 ],[ 1 ] ],[ [ 3, 2, 1 ],[ ] ] ]

See also LittlewoodRichardsonRule LittlewoodRichardsonRule.

This function requires the package ``specht'' (see RequirePackage).

71.39 EResidueDiagram

EResidueDiagram(H|e, mu)
EResidueDiagram(x)

The e--residue of the (i,j)--th node in the diagram of a partition mu is (j-i)bmod <e>. EResidueDiagram(e, mu) prints the diagram of the partition mu replacing each node with its e-residue.

If x is a module then EResidueDiagram(x) prints the e--residue diagrams of all of the e--regular partitions appearing in x (such diagrams are useful when trying to decide how to restrict and induce modules and also in applying results such as the ``Scattering theorem'' of [JM1]). It is not necessary to supply the integer e in this case because x ``knows'' the value of e.

gap> H:=Specht(2);; EResidueDiagram(H.S(H.P(7,5)));
[ 7, 5 ]
   0   1   0   1   0   1   0
   1   0   1   0   1
[ 6, 5, 1 ]
   0   1   0   1   0   1
   1   0   1   0   1
   0
[ 5, 4, 2, 1 ]
   0   1   0   1   0
   1   0   1   0
   0   1
   1
# There are 3 2-regular partitions.

This function requires the package ``specht'' (see RequirePackage).

71.40 HookLengthDiagram

HookLengthDiagram(mu)

Prints the diagram of mu, replacing each node with its hook length (see [JK]).

gap> HookLengthDiagram(11,6,3,2);
 14 13 11  9  8  7  5  4  3  2  1
  8  7  5  3  2  1
  4  3  1
  2  1

This function requires the package ``specht'' (see RequirePackage).

71.41 RemoveRimHook

RemoveRimHook(mu, row, col)

Returns the partition obtained from mu by removing the (row, col)--th rim hook from (the diagram of) mu.

gap> RemoveRimHook([6,5,4],1,2);
[ 4, 3, 1 ]
gap> RemoveRimHook([6,5,4],2,3);
[ 6, 3, 2 ]
gap> HookLengthDiagram(6,5,4);
   8   7   6   5   3   1
   6   5   4   3   1
   4   3   2   1 

See also AddRimHook AddRimHook. This function requires the package ``specht'' (see RequirePackage).

71.42 AddRimHook

AddRimHook(mu, r, h);

Returns a list [nu, l] where nu is the partition obtained from mu by adding a rim hook of length h with its ``foot'' in the r--th row of (the diagram of) mu and l is the leg length of the wrapped on rim hook (see, for example, [JK]). If the resulting diagram nu is not the diagram of a partition then false is returned.

gap> AddRimHook([6,4,3],1,3);
[ [ 9, 4, 3 ], 0 ]
gap> AddRimHook([6,4,3],2,3);
false
gap> AddRimHook([6,4,3],3,3);
[ [ 6, 5, 5 ], 1 ]
gap> AddRimHook([6,4,3],4,3);
[ [ 6, 4, 3, 3 ], 0 ]
gap> AddRimHook([6,4,3],5,3);
false

See also RemoveRimHook RemoveRimHook. This function requires the package ``specht'' (see RequirePackage).

71.43 Operations on partitions

This section contains functions for manipulating partitions and also several useful orderings on the set of partitions.

71.44 ECore

ECore(H|e, mu)

The e-core of a partition mu is what remains after as many rim e-hooks as possible have been removed from the diagram of mu (that this is well defined is not obvious; see [JK]). Thus, ECore(mu) returns the e--core of the partition mu,

gap> H:=Specht(6);; ECore(H,16,8,6,5,3,1);
[ 4, 3, 1, 1 ]

The e--core is calculated here using James' notation of an abacus; there is also an EAbacus function; but it is more ``pretty'' than useful.

See also IsECore IsECore, EQuotient EQuotient, and EWeight EWeight. This function requires the package ``specht'' (see RequirePackage).

71.45 IsECore

IsECore(H|e, mu)

Returns true if mu is an e--core and false otherwise; see ECore ECore.

See also ECore ECore. This function requires the package ``specht'' (see RequirePackage).

71.46 EQuotient

EQuotient(H|e, mu)

Returns the e-quotient of mu; this is a sequence of e partitions whose definition can be found in [JK].

gap> H:=Specht(8);; EQuotient(H,22,18,16,12,12,1,1);
[ [ 1, 1 ], [  ], [  ], [  ], [  ], [ 2, 2 ], [  ], [ 1 ] ]

See also ECore ECore and CombineEQuotientECore CombineEQuotientECore. This function requires the package ``specht'' (see RequirePackage).

71.47 CombineEQuotientECore

CombineEQuotientECore(H|e, Q, C)

A partition is uniquely determined by its e-quotient and its e-core (see EQuotient and ECore). CombineEQuotientECore(e, Q, C) returns the partition which has e--quotient Q and e--core C. The integer e can be replaced with a record H which was created using the function Specht.

gap> H:=Specht(11);; mu:=[100,98,57,43,12,1];;
gap> Q:=EQuotient(H,mu);
[ [ 9 ], [  ], [  ], [  ], [  ], [  ], [ 3 ], [ 1 ], [ 9 ], [  ], [ 5 ] ]
gap> C:=ECore(H,mu);
[ 7, 2, 2, 1, 1, 1 ]
gap> CombineEQuotientECore(H,Q,C);
[ 100, 98, 57, 43, 12, 1 ]

See also ECore ECore and EQuotient EQuotient. This function requires the package ``specht'' (see RequirePackage).

71.48 EWeight

EWeight(H|e, mu)

The e--weight of a partition is the number of e--hooks which must be removed from the partition to reach the e--core (see ECore ECore).

gap> EWeight(6,[16,8,6,5,3,1]);
5

This function requires the package ``specht'' (see RequirePackage).

71.49 ERegularPartitions

ERegularPartitions(H|e, n)

A partition mu=(mu_1,mu_2,ldots) is e--regular if there is no integer i such that mu_i=mu_{i+1}=cdots=mu_{i+<e>-1}>0. The function ERegularPartitions(e, n) returns the list of e--regular partitions of n, ordered reverse lexicographically (see Lexicographic).

gap> H:=Specht(3);
Specht(e=3, S(), P(), D(), Pq());
gap> ERegularPartitions(H,6);       
[ [ 2, 2, 1, 1 ], [ 3, 2, 1 ], [ 3, 3 ], [ 4, 1, 1 ], [ 4, 2 ], 
  [ 5, 1 ], [ 6 ] ] 

This function requires the package ``specht'' (see RequirePackage).

71.50 IsERegular

IsERegular(H|e, mu)

Returns true if mu is e--regular and false otherwise.

This functions requires the package ``specht'' (see RequirePackage).

71.51 ConjugatePartition

ConjugatePartition(mu)

Given a partition mu, ConjugatePartition(mu) returns the partition whose diagram is obtained by interchanging the rows and columns in the diagram of mu.

gap> ConjugatePartition(6,4,3,2);
[ 4, 4, 3, 2, 1, 1 ]

This function requires the package ``specht'' (see RequirePackage).

71.52 ETopLadder

ETopLadder(H|e, mu)

The ladders in the diagram of a partition are the lines connecting nodes of constant e--residue, having slope <e>-1 (see [JK]). A new partition can be obtained from mu by sliding all nodes up to the highest possible rungs on their ladders. ETopLadder(e, mu) returns the partition obtained in this way; it is automatically e--regular (this partition is denoted mu^R in [JK]).

gap> H:=Specht(4);;
gap> ETopLadder(H,1,1,1,1,1,1,1,1,1,1);
[ 4, 3, 3 ]
gap> ETopLadder(6,1,1,1,1,1,1,1,1,1,1); 
[ 2, 2, 2, 2, 2 ]

This function requires the package ``specht'' (see RequirePackage).

71.53 Dominates

Dominates(mu, nu)

The dominance ordering is an important partial order in the representation theory of Hecke algebra because d_{munu}=0 unless nu dominates mu. Dominates(mu, nu) returns true if either mu=nu or for all i ge 1, sum_{j=1}^imu_j ge sum_{j=1}^inu_j, and false otherwise.

gap> Dominates([5,4],[4,4,1]);
true

This function requires the package ``specht'' (see RequirePackage).

71.54 LengthLexicographic

LengthLexicographic(mu, nu)

LengthLexicographic returns true if the length of mu is less than the length of nu or if the length of mu equals the length of nu and Lexicographic(mu, nu).

gap> p:=Partitions(6);;Sort(p,LengthLexicographic); p;
[ [ 6 ],[ 5, 1 ],[ 4, 2 ],[ 3, 3 ],[ 4, 1, 1 ],[ 3, 2, 1 ],[ 2, 2, 2 ],
  [ 3, 1, 1, 1 ],[ 2, 2, 1, 1 ],[ 2, 1, 1, 1, 1 ],[ 1, 1, 1, 1, 1, 1 ] ]

This function requires the package ``specht'' (see RequirePackage).

71.55 Lexicographic

Lexicographic(mu, nu)

Lexicographic(mu, nu) returns true if mu is lexicographically greater than or equal to nu.

gap> p:=Partitions(6);;Sort(p,Lexicographic); p;
[ [ 6 ],[ 5, 1 ],[ 4, 2 ],[ 4, 1, 1 ],[ 3, 3 ],[ 3, 2, 1 ],
  [ 3, 1, 1, 1 ],[ 2, 2, 2 ],[ 2, 2, 1, 1 ],[ 2, 1, 1, 1, 1 ],
  [ 1, 1, 1, 1, 1, 1 ] ]

This function requires the package ``specht'' (see RequirePackage).

71.56 ReverseDominance

ReverseDominance(mu, nu)

This is another total order on partitions which extends the dominance ordering (see Dominates). Here mu is greater than nu if for all i>0 sum_jge imu_j sum_jge inu_j.

gap> p:=Partitions(6);;Sort(p,ReverseDominance); p;
[ [ 6 ], [ 5, 1 ], [ 4, 2 ], [ 3, 3 ], [ 4, 1, 1 ], [ 3, 2, 1 ], 
  [ 2, 2, 2 ], [ 3, 1, 1, 1 ], [ 2, 2, 1, 1 ], [ 2, 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1, 1, 1 ] ] 

This is the ordering used by James in the appendix of his Springer lecture notes book.

This function requires the package ``specht'' (see RequirePackage).

71.57 Miscellaneous functions on modules

This section contains some functions for looking at the partitions in a given module for the Hecke algebras. Most of them are used internally by Specht.

71.58 Specialized

Specialized(x [,q]);
Specialized(d [,q]);

Given an element of the Fock space x (see Specht), or a crystallized decomposition matrix (see CrystalDecompositionMatrix), Specialized returns the corresponding element of the Grothendieck ring or the corresponding decomposition matrix of the Hecke algebra respectively. By default the indeterminate v is specialized to 1; however v can be specialized to any (integer) q by supplying a second argument.

gap> H:=Specht(2);; x:=H.Pq(6,2);
S(6,2)+v*S(6,1,1)+v*S(5,3)+v^2*S(5,1,1,1)+v*S(4,3,1)+v^2*S(4,2,2)
+(v^3 + v)*S(4,2,1,1)+v^2*S(4,1,1,1,1)+v^2*S(3,3,1,1)+v^3*S(3,2,2,1)
+v^3*S(3,1,1,1,1,1)+v^3*S(2,2,2,1,1)+v^4*S(2,2,1,1,1,1)
gap> Specialized(x);
S(6,2)+S(6,1,1)+S(5,3)+S(5,1,1,1)+S(4,3,1)+S(4,2,2)
+2*S(4,2,1,1)+S(4,1,1,1,1)+S(3,3,1,1)+S(3,2,2,1)+S(3,1,1,1,1,1)
+S(2,2,2,1,1)+S(2,2,1,1,1,1)
gap> Specialized(x,2);
S(6,2)+2*S(6,1,1)+2*S(5,3)+4*S(5,1,1,1)+2*S(4,3,1)+4*S(4,2,2)+10*S(4,2,1,1)
+4*S(4,1,1,1,1)+4*S(3,3,1,1)+8*S(3,2,2,1)+8*S(3,1,1,1,1,1)+8*S(2,2,2,1,1)
+16*S(2,2,1,1,1,1) 

An example of Specialize being applied to a crystallized decomposition matrix can be found in CrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

71.59 ERegulars

ERegulars(x)
ERegulars(d)
ListERegulars(x)

ERegulars(x) prints a list of the e--regular partitions, together with multiplicities, which occur in the module x. ListERegulars(x) returns an actual list of these partitions rather than printing them.

gap> H:=Specht(8);;
gap> x:=H.S(InducedModule(H.P(8,5,3)) );  
S(9,5,3)+S(8,6,3)+S(8,5,4)+S(8,5,3,1)+S(6,5,3,3)+S(5,5,4,3)+S(5,5,3,3,1)
gap> ERegulars(x);
[ 9, 5, 3 ]  [ 8, 6, 3 ]  [ 8, 5, 4 ]  [ 8, 5, 3, 1 ]  
[ 6, 5, 3, 3 ]  [ 5, 5, 4, 3 ]  [ 5, 5, 3, 3, 1 ]  
gap> H.P(x);
P(9,5,3)+P(8,6,3)+P(8,5,4)+P(8,5,3,1)

This example shows why these functions are useful: given a projective module x, as above, and the list of e--regular partitions in x we know the possible indecomposable direct summands of x.

Note that it is not necessary to specify what e is when calling this function because x ``knows'' the value of e.

The function ERegulars can also be applied to a decomposition matrix d; in this case it returns the unitriangular submatrix of d whose rows and columns are indexed by the e--regular partitions.

These function requires the package ``specht'' (see RequirePackage).

71.60 SplitECores

SplitECores(x)
SplitECores(x, mu)
SplitECores(x, y)

The function SplitECores(x) returns a list [b_1,...,b_k] where the Specht modules in each b_i all belong to the same block (ie. they have the same e-core). Similarly, SplitECores(x, mu) returns the component of x which is in the same block as mu, and SplitECores(x, y) returns the component of x which is in the same block as y.

gap> H:=Specht(2);;
gap> SplitECores(InducedModule(H.S(5,3,1))); 
[ S(6,3,1)+S(5,3,2)+S(5,3,1,1), S(5,4,1) ]
gap> InducedModule(H.S(5,3,1),0);
S(5,4,1)
gap> InducedModule(H.S(5,3,1),1);
S(6,3,1)+S(5,3,2)+S(5,3,1,1) 

See also ECore ECore, InducedModule InducedModule, and RestrictedModule RestrictedModule.

This function requires the package ``specht'' (see RequirePackage).

71.61 Coefficient for Sums of Modules

Coefficient(x, mu)

If x is a sum of Specht (resp. simple, or indecomposable) modules, then Coefficient(x, mu) returns the coefficient of S(mu) in x (resp. D(mu), or P(mu)).

gap> H:=Specht(3);; x:=H.S(H.P(7,3));
S(7,3)+S(7,2,1)+S(6,2,1^2)+S(5^2)+S(5,2^2,1)+S(4^2,1^2)+S(4,3^2)+S(4,3,2,1)
gap> Coefficient(x,5,2,2,1);  
1

This function requires the package ``specht'' (see RequirePackage).

71.62 InnerProduct

InnerProduct(x, y)

Here x and y are some modules of the Hecke algebra (ie. Specht modules, PIMS, or simple modules). InnerProduct(x, y) computes the standard inner product of these elements. This is sometimes a convenient way to compute decomposition numbers (for example).

gap> InnerProduct(H.S(2,2,2,1), H.P(4,3));
1
gap> DecompositionNumber(H,[2,2,2,1],[4,3]);
1 

This function requires the package ``specht'' (see RequirePackage).

71.63 SpechtPrettyPrint

SpechtPrettyPrint(true)
SpechtPrettyPrint(false)
SpechtPrettyPrint()

This function changes the way in which Specht prints modules. The first two forms turn pretty printing on and off respectively (by default it is off), and the third form toggles the printing format.

gap> H:=Specht(2);; x:=H.S(H.P(6));;
gap> SpechtPrettyPrint(true); x;
S(6)+S(5,1)+S(4,1^2)+S(3,1^3)+S(2,1^4)+S(1^6)
gap> SpechtPrettyPrint(false); x;
S(6)+S(5,1)+S(4,1,1)+S(3,1,1,1)+S(2,1,1,1,1)+S(1,1,1,1,1,1)
gap> SpechtPrettyPrint(); x;     
S(6)+S(5,1)+S(4,1^2)+S(3,1^3)+S(2,1^4)+S(1^6)

This function requires the package ``specht'' (see RequirePackage).

71.64 Semi--standard and standard tableaux

These functions are not really part of Specht proper; however they are related and may well be of use to someone. Tableaux are represented as lists, where the first element of the list is the first row of the tableaux and so on.

71.65 SemiStandardTableaux

SemiStandardTableaux(mu, nu)

mu a partition, nu a composition.

Returns a list of the semistandard mu--tableaux of type nu [JK]. Tableaux are represented as lists of lists, with the first element of the list being the first row of the tableaux and so on.

gap> SemiStandardTableaux([4,3],[1,1,1,2,2]);    
[ [ [ 1, 2, 3, 4 ], [ 4, 5, 5 ] ], [ [ 1, 2, 3, 5 ], [ 4, 4, 5 ] ], 
  [ [ 1, 2, 4, 4 ], [ 3, 5, 5 ] ], [ [ 1, 2, 4, 5 ], [ 3, 4, 5 ] ], 
  [ [ 1, 3, 4, 4 ], [ 2, 5, 5 ] ], [ [ 1, 3, 4, 5 ], [ 2, 4, 5 ] ] ]

See also StandardTableaux StandardTableaux. This function requires the package ``specht'' (see RequirePackage).

71.66 StandardTableaux

StandardTableaux(mu)

mu a partition.

Returns a list of the standard mu--tableaux.

gap> StandardTableaux(4,2);
[ [ [ 1, 2, 3, 4 ], [ 5, 6 ] ], [ [ 1, 2, 3, 5 ], [ 4, 6 ] ], 
  [ [ 1, 2, 3, 6 ], [ 4, 5 ] ], [ [ 1, 2, 4, 5 ], [ 3, 6 ] ], 
  [ [ 1, 2, 4, 6 ], [ 3, 5 ] ], [ [ 1, 2, 5, 6 ], [ 3, 4 ] ], 
  [ [ 1, 3, 4, 5 ], [ 2, 6 ] ], [ [ 1, 3, 4, 6 ], [ 2, 5 ] ], 
  [ [ 1, 3, 5, 6 ], [ 2, 4 ] ] ]

See also SemiStandardTableaux SemiStandardTableaux. This function requires the package ``specht'' (see RequirePackage).

71.67 ConjugateTableau

ConjugateTableau(tab)

Returns the tableau obtained from tab by interchangings its rows and columns.

gap> ConjugateTableau([ [ 1, 3, 5, 6 ], [ 2, 4 ] ]);
[ [ 1, 2 ], [ 3, 4 ], [ 5 ], [ 6 ] ] 

This function requires the package ``specht'' (see RequirePackage).

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GAP 3.4.4
April 1997