Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.

Revised:

Accepted:

Published online:

Classification: 05E10, 20C30, 20G05

Keywords: character polynomial, restriction problem.

@article{ALCO_2021__4_4_703_0, author = {Narayanan, Sridhar P. and Paul, Digjoy and Prasad, Amritanshu and Srivastava, Shraddha}, title = {Character {Polynomials} and the {Restriction} {Problem}}, journal = {Algebraic Combinatorics}, pages = {703--722}, publisher = {MathOA foundation}, volume = {4}, number = {4}, year = {2021}, doi = {10.5802/alco.176}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.176/} }

Narayanan, Sridhar P.; Paul, Digjoy; Prasad, Amritanshu; Srivastava, Shraddha. Character Polynomials and the Restriction Problem. Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 703-722. doi : 10.5802/alco.176. https://alco.centre-mersenne.org/articles/10.5802/alco.176/

[1] Schur functors and Schur complexes, Adv. in Math., Volume 44 (1982) no. 3, pp. 207-278 | Article | MR 658729 | Zbl 0497.15020

[2] Specht modules decompose as alternating sums of restrictions of Schur modules, Proc. Amer. Math. Soc., Volume 148 (2020) no. 3, pp. 1015-1029 | Article | MR 4055931 | Zbl 1455.20007

[3] FI-modules and stability for representations of symmetric groups, Duke Math. J., Volume 164 (2015) no. 9, pp. 1833-1910 | Article | MR 3357185 | Zbl 1339.55004

[4] FI-modules over Noetherian rings, Geom. Topol., Volume 18 (2014) no. 5, pp. 2951-2984 | Article | MR 3285226 | Zbl 1344.20016

[5] Cohomology of finite group schemes over a field, Invent. Math., Volume 127 (1997) no. 2, pp. 209-270 | Article | MR 1427618 | Zbl 0945.14028

[6] Character polynomials, their $q$-analogs and the Kronecker product, Electron. J. Combin., Volume 16 (2009) no. 2, Special volume in honor of Anders Björner, 40 pages | Article | MR 2576382 | Zbl 1216.20003

[7] Polynomial representations of ${\mathrm{GL}}_{n}$, Lecture Notes in Mathematics, 830, Springer, Berlin Heidelberg, 2007 | MR 2349209

[8] Representations of monomial matrices and restriction from ${GL}_{n}$ to ${S}_{n}$ (2018) (https://arxiv.org/abs/1804.04702)

[9] The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass., 1981 | MR 644144 | Zbl 0491.20010

[10] Partitions of bipartite numbers, Graphs Combin., Volume 13 (1997) no. 1, pp. 73-78 | Article | MR 1435658 | Zbl 0869.11079

[11] Schur and Weyl functors, Adv. Math., Volume 90 (1991) no. 1, pp. 77-113 | Article | MR 1133476 | Zbl 0817.20049

[12] Koszul, Ringel and Serre duality for strict polynomial functors, Compos. Math., Volume 149 (2013) no. 6, pp. 996-1018 | Article | MR 3077659 | Zbl 1293.20046

[13] Combinatorics: the Rota way, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2009, xii+396 pages | Article | MR 2483561 | Zbl 1159.05002

[14] Partitions of bi-partite numbers into at most $j$ parts, Graphs Combin., Volume 8 (1992) no. 1, pp. 65-73 | Article | MR 1157510 | Zbl 0762.05005

[15] Products and plethysms of characters with orthogonal, symplectic and symmetric groups, Canadian J. Math., Volume 10 (1958), pp. 17-32 | Article | MR 95209 | Zbl 0079.03604

[16] Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, xii+475 pages | MR 3443860

[17] Polynomial induction and the restriction problem (2020) (To appear in the proceedings of Group Algebras, Representations and Computation, held at ICTS Bangalore in 2019 in a special issue of Indian J. Pure Appl. Math., https://arxiv.org/abs/2004.03928)

[18] The Hopf structure of symmetric group characters as symmetric functions (2018) (https://arxiv.org/abs/1901.00378) | Zbl 07367707

[19] Symmetric group characters as symmetric functions (2018) (https://arxiv.org/abs/1605.06672v4)

[20] Representation theory: A combinatorial viewpoint, Cambridge Studies in Advanced Mathematics, 147, Cambridge University Press, Delhi, 2015, xii+191 pages | Article | MR 3287258 | Zbl 1331.20007

[21] A Hopf-algebra approach to inner plethysm, Adv. Math., Volume 104 (1994) no. 1, pp. 30-58 | Article | MR 1272068 | Zbl 0830.20027

[22] Lectures on bifunctors and finite generation of rational cohomology algebras, Lectures on functor homology (Progr. Math.), Volume 311, Birkhäuser/Springer, Cham, 2015, pp. 41-65 | Article | MR 3468191 | Zbl 1347.20052