Splitting Permutation Representations of Finite Groups by Polynomial Algebra Methods

Abstract

An algorithm for splitting permutation representations of a finite group over fields of characteristic zero into irreducible components is described. The algorithm is based on the fact that the components of the invariant inner product in invariant subspaces are operators of projection into these subspaces. An important part of the algorithm is the solution of systems of quadratic equations. A preliminary implementation of the algorithm splits representations up to dimensions of hundreds of thousands. Examples of computations are given in the appendix.

A Examples of Computations

• Generators of representations are taken from the section “Sporadic groups” of the Atlas [7].

• For a group

  • denotes the Schur multiplier, the nd homology group ,

  • denotes the outer automorphism group of ,

  • denotes a covering group of , a central extension of by .

• The results presented below assume the following ordering for the centralizer ring basis matrices

  

  The matrices within the first sublist are ordered by the rule: if , where . The same rule is applied to the first elements of the pairs of asymmetric matrices.

• Representations are denoted by their dimensions in bold (possibly with some signs added to distinguish different representations of the same dimension). Permutation representations are underlined. Multiple subrepresentations are underbraced in the decompositions.

• We omit the irreducible projectors related to the trivial subrepresentation: these projectors have the standard form .

• All timing data refer to a PC with 3.30 GHz Intel Core i3 2120 CPU.

1.1 A.1 Higman–Sims Group

Main properties:

$\varvec{11200}$-dimensional Representation of $\varvec{2.HS}$ 

Rank: . Suborbit lengths:

Time C: s. Time Maple: h min s.

1.2 A.2 Held Group

Main properties:

 

$\varvec{29155}$-dimensional Representation of ${\varvec{He}}$ 

Rank: . Suborbit lengths: .

$$\begin{aligned} \mathcal {B}_{\mathbf {4352}} =&\frac{256}{1715}\left( \mathcal {A}_{1}+\frac{1}{8}\mathcal {A}_{2}+\frac{7}{768}\mathcal {A}_{3}-\frac{5}{576}\mathcal {A}_{4}-\frac{7}{128}\mathcal {A}_{6}-\frac{1}{48}\mathcal {A}_{7} \right. \\ {}&\left. \qquad \quad -\frac{1}{18}\mathcal {A}_{8}+\frac{1}{576}\mathcal {A}_{9}+\frac{1}{576}\mathcal {A}_{10}-\frac{1}{192}\mathcal {A}_{11}-\frac{1}{192}\mathcal {A}_{12} \right) \\ \mathcal {B}_{\mathbf {7650}} =&\frac{90}{343}\left( \mathcal {A}_{1}-\frac{1}{20}\mathcal {A}_{2}+\frac{1}{120}\mathcal {A}_{4}-\frac{7}{360}\mathcal {A}_{5}-\frac{1}{90}\mathcal {A}_{7}+\frac{1}{10}\mathcal {A}_{8}+\frac{1}{240}\mathcal {A}_{9} \right. \\ {}&\left. \qquad \quad +\frac{1}{240}\mathcal {A}_{10}-\frac{1}{80}\mathcal {A}_{11}-\frac{1}{80}\mathcal {A}_{12} \right) \\ \mathcal {B}_{\mathbf {11900}} =&\frac{20}{49}\left( \mathcal {A}_{1}-\frac{1}{20}\mathcal {A}_{2}-\frac{1}{720}\mathcal {A}_{4}+\frac{1}{120}\mathcal {A}_{7} -\frac{1}{18}\mathcal {A}_{8}-\frac{1}{180}\mathcal {A}_{9}-\frac{1}{180}\mathcal {A}_{10} \right. \\ {}&\left. \qquad \quad +\frac{1}{60}\mathcal {A}_{11}+\frac{1}{60}\mathcal {A}_{12} \right) \end{aligned}$$

Time C: s. Time Maple: s.

1.3 A.3 Suzuki Group

Main properties:

 

$\varvec{65520}$-dimensional Representation of $\varvec{2.Suz}$ 

Rank: . Suborbit lengths: .

$$\begin{aligned}&\mathbf {\underline{65520}}\cong \mathbf {1}\oplus \mathbf {143}\oplus \mathbf {364_\alpha }\oplus \mathbf {364_\beta }\oplus \overline{\mathbf {364_\beta }}\oplus \mathbf {5940}\oplus \mathbf {12012}\oplus \mathbf {14300}\\&\quad \quad \quad \quad \,\,\,\,\, \oplus \mathbf {16016}\oplus \overline{\mathbf {16016}} \end{aligned}$$

$$\begin{aligned} \mathcal {B}_{\mathbf {143}} =&\frac{11}{5040}\left( \mathcal {A}_{1}+\mathcal {A}_{2}+\frac{2}{11}\mathcal {A}_{3}-\frac{1}{11}\mathcal {A}_{4}+\frac{2}{11}\mathcal {A}_{5}-\frac{1}{11}\mathcal {A}_{6}+\frac{3}{11}\mathcal {A}_{9}+\frac{3}{11}\mathcal {A}_{10} \right) \\ \mathcal {B}_{\mathbf {364_\alpha }} =&\frac{1}{180}\left( \mathcal {A}_{1}+\mathcal {A}_{2}+\frac{1}{16}\mathcal {A}_{3}+\frac{1}{6}\mathcal {A}_{4}+\frac{1}{16}\mathcal {A}_{5}-\frac{1}{24}\mathcal {A}_{6}-\frac{1}{144}\mathcal {A}_{7}-\frac{1}{144}\mathcal {A}_{8} \right. \\ {}&\left. \qquad \quad -\frac{1}{9}\mathcal {A}_{9}-\frac{1}{9}\mathcal {A}_{10} \right) \\ \mathcal {B}_{\mathbf {364_\beta }} =&\frac{1}{180}\left( \mathcal {A}_{1}-\mathcal {A}_{2}-\frac{1}{8}\mathcal {A}_{3}+\frac{1}{8}\mathcal {A}_{5}+\mathrm {\mathbf {i}}\frac{\sqrt{3}}{72}\mathcal {A}_{7}-\mathrm {\mathbf {i}}\frac{\sqrt{3}}{72}\mathcal {A}_{8} \right. \\ {}&\left. \qquad \quad +\mathrm {\mathbf {i}}\frac{\sqrt{3}}{9}\mathcal {A}_{9}-\mathrm {\mathbf {i}}\frac{\sqrt{3}}{9}\mathcal {A}_{10} \right) \\ \mathcal {B}_{\mathbf {5940}} =&\frac{33}{364}\left( \mathcal {A}_{1}+\mathcal {A}_{2}+\frac{1}{352}\mathcal {A}_{3}+\frac{1}{66}\mathcal {A}_{4}+\frac{1}{352}\mathcal {A}_{5}+\frac{1}{66}\mathcal {A}_{6}-\frac{7}{864}\mathcal {A}_{7} \right. \\ {}&\left. \qquad \quad -\frac{7}{864}\mathcal {A}_{8}+\frac{1}{27}\mathcal {A}_{9}+\frac{1}{27}\mathcal {A}_{10} \right) \\ \end{aligned}$$

$$\begin{aligned} \mathcal {B}_{\mathbf {12012}} =&\frac{11}{60}\left( \mathcal {A}_{1}+\mathcal {A}_{2}+\frac{1}{88}\mathcal {A}_{3}-\frac{1}{66}\mathcal {A}_{4}+\frac{1}{88}\mathcal {A}_{5}+\frac{1}{264}\mathcal {A}_{6}-\frac{1}{33}\mathcal {A}_{9}-\frac{1}{33}\mathcal {A}_{10} \right) \\ \mathcal {B}_{\mathbf {14300}} =&\frac{55}{252}\left( \mathcal {A}_{1}+\mathcal {A}_{2}-\frac{5}{352}\mathcal {A}_{3}+\frac{1}{330}\mathcal {A}_{4}-\frac{5}{352}\mathcal {A}_{5}-\frac{1}{132}\mathcal {A}_{6}+\frac{1}{288}\mathcal {A}_{7} \right. \\ {}&\left. \qquad \quad +\frac{1}{288}\mathcal {A}_{8}+\frac{1}{99}\mathcal {A}_{9}+\frac{1}{99}\mathcal {A}_{10} \right) \\ \mathcal {B}_{\mathbf {16016}} =&\frac{11}{45}\left( \mathcal {A}_{1}-\mathcal {A}_{2}+\frac{1}{352}\mathcal {A}_{3}-\frac{1}{352}\mathcal {A}_{5}-\mathrm {\mathbf {i}}\frac{\sqrt{3}}{288}\mathcal {A}_{7}+\mathrm {\mathbf {i}}\frac{\sqrt{3}}{288}\mathcal {A}_{8} \right. \\ {}&\left. \qquad \quad +\mathrm {\mathbf {i}}\frac{\sqrt{3}}{99}\mathcal {A}_{9}-\mathrm {\mathbf {i}}\frac{\sqrt{3}}{99}\mathcal {A}_{10} \right) \end{aligned}$$

Time C: min s. Time Maple: s.

$\varvec{98280}$-dimensional Representation of $\varvec{3.Suz}$ 

Rank: . Suborbit lengths: .

$$\begin{aligned}&\mathbf {\underline{98280}}\cong \mathbf {1}\oplus \mathbf {78}\oplus \overline{\mathbf {78}}\oplus \mathbf {143}\oplus \mathbf {364}\oplus \mathbf {1365}\oplus \overline{\mathbf {1365}}\oplus \mathbf {4290}\oplus \overline{\mathbf {4290}}\\&\quad \quad \quad \,\,\,\;\;\, \oplus \mathbf {5940}\oplus \mathbf {12012}\oplus \mathbf {14300}\oplus \mathbf {27027}\oplus \overline{\mathbf {27027}} \end{aligned}$$

$$\begin{aligned} \mathcal {B}_{\mathbf {5940}} =&\frac{11}{182}\left( \mathcal {A}_{1}-\frac{7}{864}\mathcal {A}_{2}+\frac{1}{66}\mathcal {A}_{3}+\frac{1}{27}\mathcal {A}_{4}+\frac{1}{66}\mathcal {A}_{5}+\frac{1}{352}\mathcal {A}_{6}-\frac{7}{864}\mathcal {A}_{7} \right. \\ {}&\left. \quad \quad -\frac{7}{864}\mathcal {A}_{8}+\frac{1}{352}\mathcal {A}_{9}+\frac{1}{352}\mathcal {A}_{10}+\frac{1}{27}\mathcal {A}_{11}+\frac{1}{27}\mathcal {A}_{12}+\mathcal {A}_{13}+\mathcal {A}_{14} \right) \\ \mathcal {B}_{\mathbf {12012}} =&\frac{11}{90}\left( \mathcal {A}_{1}-\frac{1}{66}\mathcal {A}_{3}-\frac{1}{33}\mathcal {A}_{4}+\frac{1}{264}\mathcal {A}_{5}+\frac{1}{88}\mathcal {A}_{6}+\frac{1}{88}\mathcal {A}_{9} \right. \\ {}&\left. \quad \quad \; +\frac{1}{88}\mathcal {A}_{10}-\frac{1}{33}\mathcal {A}_{11}-\frac{1}{33}\mathcal {A}_{12}+\mathcal {A}_{13}+\mathcal {A}_{14} \right) \\ \mathcal {B}_{\mathbf {14300}} =&\frac{55}{378}\left( \mathcal {A}_{1}+\frac{1}{288}\mathcal {A}_{2}+\frac{1}{330}\mathcal {A}_{3}+\frac{1}{99}\mathcal {A}_{4}-\frac{1}{132}\mathcal {A}_{5}-\frac{5}{352}\mathcal {A}_{6}+\frac{1}{288}\mathcal {A}_{7} \right. \\ {}&\left. \quad \quad \; +\frac{1}{288}\mathcal {A}_{8}-\frac{5}{352}\mathcal {A}_{9}-\frac{5}{352}\mathcal {A}_{10}+\frac{1}{99}\mathcal {A}_{11}+\frac{1}{99}\mathcal {A}_{12}+\mathcal {A}_{13}+\mathcal {A}_{14} \right) \\ \mathcal {B}_{\mathbf {27027}} =&\frac{11}{40}\left( \mathcal {A}_{1}-\frac{1}{432}\mathcal {A}_{2}+\frac{1}{297}\mathcal {A}_{4}-\frac{1}{176}\mathcal {A}_{6}-\frac{\mathsf {r}_{}}{432}\mathcal {A}_{7}-\frac{\mathsf {r}_{}^2}{432}\mathcal {A}_{8} \right. \\ {}&\left. \quad \quad \quad -\frac{\mathsf {r}_{}}{176}\mathcal {A}_{9}-\frac{\mathsf {r}_{}^2}{176}\mathcal {A}_{10}+\frac{\mathsf {r}_{}^2}{297}\mathcal {A}_{11}+\frac{\mathsf {r}_{}}{297}\mathcal {A}_{12}+\mathsf {r}_{}\mathcal {A}_{13}+\mathsf {r}_{}^2\mathcal {A}_{14} \right) \end{aligned}$$

is the basic primitive rd root of unity.

Time C: min s. Time Maple: min s.