Character Polynomials and the Restriction Problem
Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 703-722.

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.

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DOI: 10.5802/alco.176
Classification: 05E10,  20C30,  20G05
Keywords: character polynomial, restriction problem.
Narayanan, Sridhar P. 1; Paul, Digjoy 2; Prasad, Amritanshu 1; Srivastava, Shraddha 3

1 The Institute of Mathematical Sciences (HBNI) CIT campus, Taramani Chennai 600041, India
2 Tata Institute of Fundamental Research Colaba, Mumbai 400005, India
3 Department of Mathematics, Uppsala University Box 480, SE75106, Uppsala, Sweden
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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] Akin, Kaan; Buchsbaum, David A.; Weyman, Jerzy Schur functors and Schur complexes, Adv. in Math., Volume 44 (1982) no. 3, pp. 207-278 | DOI | MR | Zbl

[2] Assaf, Sami H.; Speyer, David E. Specht modules decompose as alternating sums of restrictions of Schur modules, Proc. Amer. Math. Soc., Volume 148 (2020) no. 3, pp. 1015-1029 | DOI | MR | Zbl

[3] Church, Thomas; Ellenberg, Jordan S.; Farb, Benson FI-modules and stability for representations of symmetric groups, Duke Math. J., Volume 164 (2015) no. 9, pp. 1833-1910 | DOI | MR | Zbl

[4] Church, Thomas; Ellenberg, Jordan S.; Farb, Benson; Nagpal, Rohit FI-modules over Noetherian rings, Geom. Topol., Volume 18 (2014) no. 5, pp. 2951-2984 | DOI | MR | Zbl

[5] Friedlander, Eric M.; Suslin, Andrei Cohomology of finite group schemes over a field, Invent. Math., Volume 127 (1997) no. 2, pp. 209-270 | DOI | MR | Zbl

[6] Garsia, Adriano M.; Goupil, Alain 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 | DOI | MR | Zbl

[7] Green, James A. Polynomial representations of GL n , Lecture Notes in Mathematics, 830, Springer, Berlin Heidelberg, 2007 | MR

[8] Harman, N. Representations of monomial matrices and restriction from GL n to S n (2018) (https://arxiv.org/abs/1804.04702)

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

[10] Kim, Jun Kyo; Hahn, Sang Geun Partitions of bipartite numbers, Graphs Combin., Volume 13 (1997) no. 1, pp. 73-78 | DOI | MR | Zbl

[11] Kouwenhoven, Frank M. Schur and Weyl functors, Adv. Math., Volume 90 (1991) no. 1, pp. 77-113 | DOI | MR | Zbl

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

[13] Kung, Joseph P. S.; Rota, Gian-Carlo; Yan, Catherine H. Combinatorics: the Rota way, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2009, xii+396 pages | DOI | MR | Zbl

[14] Landman, Bruce M.; Brown, Ezra A.; Portier, Frederick J. Partitions of bi-partite numbers into at most j parts, Graphs Combin., Volume 8 (1992) no. 1, pp. 65-73 | DOI | MR | Zbl

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

[16] Macdonald, Ian G. 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

[17] Narayanan, Sridhar P.; Paul, Digjoy; Prasad, Amritanshu; Srivastava, Shraddha 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] Orellana, Rosa; Zabrocki, Mike The Hopf structure of symmetric group characters as symmetric functions (2018) (https://arxiv.org/abs/1901.00378) | Zbl

[19] Orellana, Rosa; Zabrocki, Mike Symmetric group characters as symmetric functions (2018) (https://arxiv.org/abs/1605.06672v4)

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

[21] Scharf, Thomas; Thibon, Jean-Yves A Hopf-algebra approach to inner plethysm, Adv. Math., Volume 104 (1994) no. 1, pp. 30-58 | DOI | MR | Zbl

[22] van der Kallen, Wilberd 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 | DOI | MR | Zbl

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