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.
Revised:
Accepted:
Published online:
Keywords: symmetric functions, symmetric group characters, Hopf algebra
@article{ALCO_2021__4_3_551_0, author = {Orellana, Rosa and Zabrocki, Mike}, title = {The {Hopf} structure of symmetric group characters as symmetric functions}, journal = {Algebraic Combinatorics}, pages = {551--574}, publisher = {MathOA foundation}, volume = {4}, number = {3}, year = {2021}, doi = {10.5802/alco.170}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.170/} }
TY - JOUR AU - Orellana, Rosa AU - Zabrocki, Mike TI - The Hopf structure of symmetric group characters as symmetric functions JO - Algebraic Combinatorics PY - 2021 SP - 551 EP - 574 VL - 4 IS - 3 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.170/ DO - 10.5802/alco.170 LA - en ID - ALCO_2021__4_3_551_0 ER -
%0 Journal Article %A Orellana, Rosa %A Zabrocki, Mike %T The Hopf structure of symmetric group characters as symmetric functions %J Algebraic Combinatorics %D 2021 %P 551-574 %V 4 %N 3 %I MathOA foundation %U http://www.numdam.org/articles/10.5802/alco.170/ %R 10.5802/alco.170 %G en %F ALCO_2021__4_3_551_0
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] 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] A combinatorial interpretation for the coefficients in the Kronecker product , Sém. Lothar. Combin., Volume 54A (2005/07), Art. B54Af, 29 pages | MR | Zbl
[3] The lattice permutation condition for Kronecker tableaux (2018) (arXiv preprint, https://arxiv.org/abs/1812.09175) | Zbl
[4] The partition algebra and the Kronecker coefficients, Trans. Amer. Math. Soc., Volume 367 (2015) no. 5, pp. 3647-3667 | DOI | MR | Zbl
[5] 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] The stability of the Kronecker product of Schur functions, J. Algebra, Volume 331 (2011), pp. 11-27 | DOI | MR | Zbl
[7] The symmetric group: characters, products and plethysms, J. Mathematical Phys., Volume 14 (1973), pp. 1176-1183 | DOI | MR | Zbl
[8] FI-modules and stability for representations of symmetric groups, Duke Math. J., Volume 164 (2015) no. 9, pp. 1833-1910 | DOI | MR | Zbl
[9] Representation theory and homological stability, Adv. Math., Volume 245 (2013), pp. 250-314 | DOI | MR | Zbl
[10] Deligne categories and reduced Kronecker coefficients, J. Algebraic Combin., Volume 44 (2016) no. 2, pp. 345-362 | DOI | MR | Zbl
[11] Über die Charaktere der symmetrischen Gruppe, Königliche Akademie der Wissenschaften, 1900 | DOI | MR | Zbl
[12] Shuffles of permutations and the Kronecker product, Graphs Combin., Volume 1 (1985) no. 3, pp. 217-263 | DOI | MR | Zbl
[13] Hopf algebras in combinatorics (2014), 282 pages pages (arXiv preprint, https://arxiv.org/abs/1409.8356) | DOI | MR | Zbl
[14] Symmetric Functions as Characters of Hyperoctahedral Group, Ph. D. Thesis, York University (2020)
[15] Branching rules for and the evaluation of inner plethysms, J. Mathematical Phys., Volume 15 (1974), pp. 258-267 | DOI | MR
[16] Symmetric functions (2001), 108 pages (http://www.emis.de/journals/SLC/wpapers/s68vortrag/ALCoursSf2.pdf) | DOI | MR
[17] Products and plethysms of characters with orthogonal, symplectic and symmetric groups, Canadian J. Math., Volume 10 (1958), pp. 17-32 | DOI | MR
[18] 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] The characters of the symmetric group, Amer. J. Math., Volume 59 (1937) no. 4, pp. 739-753 | DOI | MR
[20] 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] On the analysis of the Kronecker product of irreducible representations of , Proc. Nat. Acad. Sci. U.S.A., Volume 41 (1955), pp. 515-518 | DOI | MR | Zbl
[22] Restriction of the irreducible representations of to the symmetric group (2000), 5 pages (homepage preprint 2021-02-01, http://rtweb.math.kyoto-u.ac.jp/home_kyo/preprint/glntosn.pdf) | DOI | MR
[23] Characters of the symmetric group as symmetric functions (2016), 31 pages (arXiv preprint, https://arxiv.org/abs/1605.06672)
[24] Products of symmetric group characters, J. Combin. Theory Ser. A, Volume 165 (2019), pp. 299-324 | DOI | MR
[25] The symmetric group: representations, combinatorial algorithms, and symmetric functions, 203, Springer Science & Business Media, 2013 | DOI | MR | Zbl
[26] Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2008 http://combinat.sagemath.org
[27] A Hopf-algebra approach to inner plethysm, Advances in Mathematics, Volume 104 (1994) no. 1, pp. 30-58
[28] Generating functions for stable branching coefficients , and , J. Phys. A, Volume 30 (1997) no. 19, pp. 6963-6975 | DOI | MR | Zbl
[29] Die charaktere der symmetrischen gruppe, Math. Z., Volume 73 (1960), pp. 312-329 | DOI | MR | Zbl
[30] Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR | Zbl
[31] Sage: System for algebra and geometry experimentation, Acm Sigsam Bulletin, Volume 39 (2005) no. 2, pp. 61-64 | DOI | MR | Zbl
[32] Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969, vii+336 pages | DOI | MR | Zbl
[33] 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
Cited by Sources: