The Hopf structure of symmetric group characters as symmetric functions
Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 551-574.

In [24] the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis.

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DOI: 10.5802/alco.170
Classification: 05E05, 05E10
Keywords: symmetric functions, symmetric group characters, Hopf algebra
Orellana, Rosa 1; Zabrocki, Mike 2

1 Dartmouth College Mathematics Department Hanover, NH 03755, USA
2 Department of Mathematics and Statistics York University Toronto, Ontario M3J 1P3, Canada
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Orellana, Rosa; Zabrocki, Mike. The Hopf structure of symmetric group characters as symmetric functions. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 551-574. doi : 10.5802/alco.170. http://www.numdam.org/articles/10.5802/alco.170/

[1] 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

[2] Ballantine, Cristina M.; Orellana, Rosa C. A combinatorial interpretation for the coefficients in the Kronecker product s (n-p,p) *s λ , Sém. Lothar. Combin., Volume 54A (2005/07), Art. B54Af, 29 pages | MR | Zbl

[3] Bowman, Chris; De Visscher, Maud; Enyang, John The lattice permutation condition for Kronecker tableaux (2018) (arXiv preprint, https://arxiv.org/abs/1812.09175) | Zbl

[4] Bowman, Christopher; De Visscher, Maud; Orellana, Rosa The partition algebra and the Kronecker coefficients, Trans. Amer. Math. Soc., Volume 367 (2015) no. 5, pp. 3647-3667 | DOI | MR | Zbl

[5] Briand, Emmanuel; Orellana, Rosa; Rosas, Mercedes Reduced Kronecker coefficients and counter-examples to Mulmuley’s strong saturation conjecture SH, Comput. Complexity, Volume 18 (2009) no. 4, pp. 577-600 (With an appendix by Ketan Mulmuley) | DOI | MR | Zbl

[6] Briand, Emmanuel; Orellana, Rosa; Rosas, Mercedes The stability of the Kronecker product of Schur functions, J. Algebra, Volume 331 (2011), pp. 11-27 | DOI | MR | Zbl

[7] Butler, Philip H.; King, Roger C. The symmetric group: characters, products and plethysms, J. Mathematical Phys., Volume 14 (1973), pp. 1176-1183 | DOI | MR | Zbl

[8] 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

[9] Church, Thomas; Farb, Benson Representation theory and homological stability, Adv. Math., Volume 245 (2013), pp. 250-314 | DOI | MR | Zbl

[10] Entova Aizenbud, Inna Deligne categories and reduced Kronecker coefficients, J. Algebraic Combin., Volume 44 (2016) no. 2, pp. 345-362 | DOI | MR | Zbl

[11] Frobenius, Georg F. Über die Charaktere der symmetrischen Gruppe, Königliche Akademie der Wissenschaften, 1900 | DOI | MR | Zbl

[12] Garsia, Adriano M.; Remmel, Jeffrey Shuffles of permutations and the Kronecker product, Graphs Combin., Volume 1 (1985) no. 3, pp. 217-263 | DOI | MR | Zbl

[13] Grinberg, Darij; Reiner, Victor Hopf algebras in combinatorics (2014), 282 pages pages (arXiv preprint, https://arxiv.org/abs/1409.8356) | DOI | MR | Zbl

[14] Islami, Arash Symmetric Functions as Characters of Hyperoctahedral Group, Ph. D. Thesis, York University (2020)

[15] King, Roger C. Branching rules for GL(N)Σ m and the evaluation of inner plethysms, J. Mathematical Phys., Volume 15 (1974), pp. 258-267 | DOI | MR

[16] Lascoux, Alain Symmetric functions (2001), 108 pages (http://www.emis.de/journals/SLC/wpapers/s68vortrag/ALCoursSf2.pdf) | DOI | MR

[17] 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

[18] 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 (With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition [ MR1354144]) | DOI | MR | Zbl

[19] Murnaghan, Francis D. The characters of the symmetric group, Amer. J. Math., Volume 59 (1937) no. 4, pp. 739-753 | DOI | MR

[20] Murnaghan, Francis D. The analysis of the Kronecker product of irreducible representations of the symmetric group, Amer. J. Math., Volume 60 (1938) no. 3, pp. 761-784 | DOI | MR

[21] Murnaghan, Francis D. On the analysis of the Kronecker product of irreducible representations of S n , Proc. Nat. Acad. Sci. U.S.A., Volume 41 (1955), pp. 515-518 | DOI | MR | Zbl

[22] Nishiyama, Kyo Restriction of the irreducible representations of GL n to the symmetric group S n (2000), 5 pages (homepage preprint 2021-02-01, http://rtweb.math.kyoto-u.ac.jp/home_kyo/preprint/glntosn.pdf) | DOI | MR

[23] Orellana, Rosa; Zabrocki, Mike Characters of the symmetric group as symmetric functions (2016), 31 pages (arXiv preprint, https://arxiv.org/abs/1605.06672)

[24] Orellana, Rosa; Zabrocki, Mike Products of symmetric group characters, J. Combin. Theory Ser. A, Volume 165 (2019), pp. 299-324 | DOI | MR

[25] Sagan, Bruce E. The symmetric group: representations, combinatorial algorithms, and symmetric functions, 203, Springer Science & Business Media, 2013 | DOI | MR | Zbl

[26] Sage-Combinat community Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2008 http://combinat.sagemath.org

[27] Scharf, Thomas; Thibon, Jean-Yves A Hopf-algebra approach to inner plethysm, Advances in Mathematics, Volume 104 (1994) no. 1, pp. 30-58

[28] Scharf, Thomas; Thibon, Jean-Yves; Wybourne, Brian G Generating functions for stable branching coefficients U(n)S n , O(n)S n and O(n-1)S n , J. Phys. A, Volume 30 (1997) no. 19, pp. 6963-6975 | DOI | MR | Zbl

[29] Specht, Wilhelm Die charaktere der symmetrischen gruppe, Math. Z., Volume 73 (1960), pp. 312-329 | DOI | MR | Zbl

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

[31] Stein, William; Joyner, David Sage: System for algebra and geometry experimentation, Acm Sigsam Bulletin, Volume 39 (2005) no. 2, pp. 61-64 | DOI | MR | Zbl

[32] Sweedler, Moss E. Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969, vii+336 pages | DOI | MR | Zbl

[33] Young, Alfred The collected papers of Alfred Young (1873–1940), Mathematical Expositions, University of Toronto Press, Toronto, Ont., Buffalo, N. Y., 1977 no. 21, xxvii+684 pages | MR | Zbl

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