Browse issues
no. 4
Some restriction coefficients for the trivial and sign representations
Narayanan, Sridhar1; Paul, Digjoy2; Prasad, Amritanshu3; Srivastava, Shraddha4
1 Tata Institute of Fundamental Research Mumbai
2 Department of Mathematics Indian Institute of Science Bangalore
3 The Institute of Mathematical Sciences (HBNI) Chennai
4 Indian Institute of Technology Dharwad
Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 1183-1195.

We use character polynomials to obtain a positive combinatorial interpretation of the multiplicity of the sign representation in irreducible polynomial representations of GL n (ℂ) indexed by two-column and hook partitions. Our method also yields a positive combinatorial interpretation for the multiplicity of the trivial representation of S n in an irreducible polynomial representation indexed by a hook partition.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.360
Classification: 05E10, 05E05, 20C30
Keywords: plethysm, character polynomials, moments, restriction problem, sign-reversing involution

Narayanan, Sridhar 1; Paul, Digjoy 2; Prasad, Amritanshu 3; Srivastava, Shraddha 4

1 Tata Institute of Fundamental Research Mumbai
2 Department of Mathematics Indian Institute of Science Bangalore
3 The Institute of Mathematical Sciences (HBNI) Chennai
4 Indian Institute of Technology Dharwad
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2024__7_4_1183_0,
     author = {Narayanan, Sridhar and Paul, Digjoy and Prasad, Amritanshu and Srivastava, Shraddha},
     title = {Some restriction coefficients for the trivial and sign representations},
     journal = {Algebraic Combinatorics},
     pages = {1183--1195},
     publisher = {The Combinatorics Consortium},
     volume = {7},
     number = {4},
     year = {2024},
     doi = {10.5802/alco.360},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.360/}
}
TY  - JOUR
AU  - Narayanan, Sridhar
AU  - Paul, Digjoy
AU  - Prasad, Amritanshu
AU  - Srivastava, Shraddha
TI  - Some restriction coefficients for the trivial and sign representations
JO  - Algebraic Combinatorics
PY  - 2024
SP  - 1183
EP  - 1195
VL  - 7
IS  - 4
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.360/
DO  - 10.5802/alco.360
LA  - en
ID  - ALCO_2024__7_4_1183_0
ER  - 
%0 Journal Article
%A Narayanan, Sridhar
%A Paul, Digjoy
%A Prasad, Amritanshu
%A Srivastava, Shraddha
%T Some restriction coefficients for the trivial and sign representations
%J Algebraic Combinatorics
%D 2024
%P 1183-1195
%V 7
%N 4
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.360/
%R 10.5802/alco.360
%G en
%F ALCO_2024__7_4_1183_0
Narayanan, Sridhar; Paul, Digjoy; Prasad, Amritanshu; Srivastava, Shraddha. Some restriction coefficients for the trivial and sign representations. Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 1183-1195. doi : 10.5802/alco.360. https://alco.centre-mersenne.org/articles/10.5802/alco.360/

[1] Alarie-Vezina, L.; Blondeau-Fournier, O.; Desrosiers, P.; Lapointe, L.; Mathieu, P. Symmetric functions in superspace: a compendium of results and open problems (including a SageMath worksheet), 2019 | arXiv

[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] Desrosiers, P.; Lapointe, L.; Mathieu, P. Classical symmetric functions in superspace, J. Algebraic Combin., Volume 24 (2006) no. 2, pp. 209-238 | DOI | Zbl

[4] Fishel, S.; Lapointe, L.; Pinto, María E. Hopf algebra structure of symmetric and quasisymmetric functions in superspace, J. Combin. Theory Ser. A, Volume 166 (2019), pp. 144-170 | DOI | Zbl

[5] Garsia, A. M.; Goupil, A. Character polynomials, their q-analogs and the Kronecker product, Electron. J. Combin., Volume 16 (2009) no. 2, Paper no. 19, 40 pages | DOI | MR | Zbl

[6] Heaton, A.; Sriwongsa, S.; Willenbring, J. F. Branching from the General Linear Group to the Symmetric Group and the Principal Embedding, Algebr. Combin., Volume 4 (2021) no. 2, pp. 189-200 | DOI | Numdam | Zbl

[7] Kim, J.; Rhoades, B. Lefschetz Theory for Exterior Algebras and Fermionic Diagonal Coinvariants, Int. Math. Res. Not. IMRN, Volume 2022 (2020) no. 4, pp. 2906-2933 | DOI | Zbl

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

[9] Loehr, N. A.; Remmel, J. B. A computational and combinatorial exposé of plethystic calculus, J. Algebraic Combin., Volume 33 (2011) no. 2, pp. 163-198 | DOI | MR | Zbl

[10] Macdonald, I. 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

[11] Narayanan, Sridhar P.; Paul, Digjoy; Prasad, Amritanshu; Srivastava, Shraddha Character polynomials and the restriction problem, Algebr. Comb., Volume 4 (2021) no. 4, pp. 703-722 | DOI | Numdam | MR | Zbl

[12] Narayanan, Sridhar P.; Paul, Digjoy; Prasad, Amritanshu; Srivastava, Shraddha Polynomial induction and the restriction problem, Indian J. Pure Appl. Math., Volume 52 (2021) no. 3, pp. 643-651 | DOI | Zbl

[13] Orellana, R.; Saliola, F.; Schilling, A.; Zabrocki, M. Plethysm and the algebra of uniform block permutations, Algebr. Combin., Volume 5 (2022) no. 5, pp. 1165-1203 | DOI | Numdam | Zbl

[14] Orellana, Rosa; Zabrocki, Mike A combinatorial model for the decomposition of multivariate polynomial rings as S n -modules, Electron. J. Combin., Volume 27 (2020) no. 3, Paper no. 3.24, 18 pages | DOI | MR | Zbl

[15] Orellana, Rosa; Zabrocki, Mike Symmetric group characters as symmetric functions, Adv. Math., Volume 390 (2021), Paper no. 107943, 34 pages | DOI | MR | Zbl

[16] Prasad, A. The Frobenius Characteristic of Character Polynomials, J. Indian Inst. Sci., Volume 102 (2022) no. 3, pp. 947-959 | DOI | Zbl

[17] Solomon, L. Invariants of finite reflection groups, Nagoya Math. J., Volume 22 (1963), pp. 57-64 | DOI | Zbl

[18] Stanley, R. P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR

[19] Weyman, Jerzy Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, 149, Cambridge University Press, Cambridge, 2003, xiv+371 pages | DOI

Cited by Sources: