Browse issues
no. 4
Splitting Kronecker squares, 2-decomposition numbers, Catalan combinatorics, and the Saxl conjecture
Bessenrodt, Christine; Bowman, Chris1
1 Department of Mathematics University of York Heslington, York, YO10 5DD United Kingdom
Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 863-899.

This paper concerns the symmetric and anti-symmetric Kronecker products of characters of the symmetric groups. We provide new closed formulas for decomposing these products, unexpected connections with 2-modular decomposition numbers, Catalan combinatorics, and a refinement of the famous Saxl conjecture.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.294
Classification: 05A05, 05A10, 05E05, 05A15, 05E10, 05E16, 81Q30
Keywords: Symmetric tensor squares, Kronecker product, symmetric groups, character theory, decomposition numbers, Catalan combinatorics.

Bessenrodt, Christine ; Bowman, Chris 1

1 Department of Mathematics University of York Heslington, York, YO10 5DD United Kingdom
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Bessenrodt, Christine; Bowman, Chris. Splitting Kronecker squares, 2-decomposition numbers, Catalan combinatorics, and the Saxl conjecture. Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 863-899. doi : 10.5802/alco.294. https://alco.centre-mersenne.org/articles/10.5802/alco.294/

[1] 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), Paper no. B54Af, 29 pages | MR | Zbl

[2] Bessenrodt, Christine Critical classes, Kronecker products of spin characters, and the Saxl conjecture, Algebr. Comb., Volume 1 (2018) no. 3, pp. 353-369 | Numdam | MR | Zbl

[3] Bessenrodt, Christine; Behns, Christiane On the Durfee size of Kronecker products of characters of the symmetric group and its double covers, J. Algebra, Volume 280 (2004) no. 1, pp. 132-144 | DOI | MR | Zbl

[4] Bessenrodt, Christine; Bowman, Christopher Multiplicity-free Kronecker products of characters of the symmetric groups, Adv. Math., Volume 322 (2017), pp. 473-529 | DOI | MR | Zbl

[5] Bessenrodt, Christine; Kleshchev, Alexander S. On Kronecker products of spin characters of the double covers of the symmetric groups, Pacific J. Math., Volume 198 (2001) no. 2, pp. 295-305 | DOI | MR | Zbl

[6] Blasiak, Jonah Kronecker coefficients for one hook shape, Sém. Lothar. Combin., Volume 77 (2016), Paper no. B77c, 40 pages | MR | Zbl

[7] Bowman, Chris; Paget, Rowena The uniqueness of plethystic factorisation, Trans. Amer. Math. Soc., Volume 373 (2020) no. 3, pp. 1653-1666 | DOI | MR | Zbl

[8] Brown, Andrew A. H.; van Willigenburg, Stephanie; Zabrocki, Mike Expressions for Catalan Kronecker products, Pacific J. Math., Volume 248 (2010) no. 1, pp. 31-48 | DOI | MR | Zbl

[9] Bürgisser, Peter; Christandl, Matthias; Ikenmeyer, Christian Nonvanishing of Kronecker coefficients for rectangular shapes, Adv. Math., Volume 227 (2011) no. 5, pp. 2082-2091 | DOI | MR | Zbl

[10] Bürgisser, Peter; Ikenmeyer, Christian; Hüttenhain, Jesko Permanent versus determinant: not via saturations, Proc. Amer. Math. Soc., Volume 145 (2017) no. 3, pp. 1247-1258 | DOI | MR | Zbl

[11] Bürgisser, Peter; Ikenmeyer, Christian; Panova, Greta No occurrence obstructions in geometric complexity theory, J. Amer. Math. Soc., Volume 32 (2019) no. 1, pp. 163-193 | DOI | MR | Zbl

[12] Bürgisser, Peter; Landsberg, J. M.; Manivel, Laurent; Weyman, Jerzy An overview of mathematical issues arising in the geometric complexity theory approach to VP ≠ VNP , SIAM J. Comput., Volume 40 (2011) no. 4, pp. 1179-1209 | DOI | MR | Zbl

[13] Christandl, Matthias; Harrow, Aram W.; Mitchison, Graeme Nonzero Kronecker coefficients and what they tell us about spectra, Comm. Math. Phys., Volume 270 (2007) no. 3, pp. 575-585 | DOI | MR | Zbl

[14] Christandl, Matthias; Mitchison, Graeme The spectra of quantum states and the Kronecker coefficients of the symmetric group, Comm. Math. Phys., Volume 261 (2006) no. 3, pp. 789-797 | DOI | MR | Zbl

[15] Christandl, Matthias; Şahinoğlu, M. Burak; Walter, Michael Recoupling coefficients and quantum entropies, Ann. Henri Poincaré, Volume 19 (2018) no. 2, pp. 385-410 | DOI | MR | Zbl

[16] Curtis, Charles W.; Reiner, Irving Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981, xxi+819 pages (With applications to finite groups and orders.)

[17] Dvir, Yoav On the Kronecker product of S n characters, J. Algebra, Volume 154 (1993) no. 1, pp. 125-140 | DOI | MR | Zbl

[18] Feit, Walter Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967, viii+186 pages

[19] Garsia, Adriano; Wallach, Nolan; Xin, Guoce; Zabrocki, Mike Kronecker coefficients via symmetric functions and constant term identities, Internat. J. Algebra Comput., Volume 22 (2012) no. 3, Paper no. 1250022, 44 pages | MR | Zbl

[20] Gesmundo, Fulvio; Ikenmeyer, Christian; Panova, Greta Geometric complexity theory and matrix powering, Differential Geom. Appl., Volume 55 (2017), pp. 106-127 | DOI | MR | Zbl

[21] Heide, Gerhard; Saxl, Jan; Tiep, Pham Huu; Zalesski, Alexandre E. Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type, Proc. Lond. Math. Soc. (3), Volume 106 (2013) no. 4, pp. 908-930 | DOI | MR | Zbl

[22] Huppert, Bertram Character theory of finite groups, De Gruyter Expositions in Mathematics, 25, Walter de Gruyter & Co., Berlin, 1998, vi+618 pages

[23] Ikenmeyer, Christian The Saxl conjecture and the dominance order, Discrete Math., Volume 338 (2015) no. 11, pp. 1970-1975 | DOI | MR | Zbl

[24] Ikenmeyer, Christian; Panova, Greta Rectangular Kronecker coefficients and plethysms in geometric complexity theory, Adv. Math., Volume 319 (2017), pp. 40-66 | DOI | MR | Zbl

[25] Isaacs, I. Martin Character theory of finite groups, Pure and Applied Mathematics, 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976, xii+303 pages | MR

[26] James, G. D. The representation theory of the symmetric groups, Lecture Notes in Mathematics, 682, Springer, Berlin, 1978, v+156 pages

[27] 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, xxviii+510 pages (With a foreword by P. M. Cohn., With an introduction by Gilbert de B. Robinson.)

[28] Liu, Ricky Ini A simplified Kronecker rule for one hook shape, Proc. Amer. Math. Soc., Volume 145 (2017) no. 9, pp. 3657-3664 | MR | Zbl

[29] Magaard, Kay On the irreducibility of alternating powers and symmetric squares, Arch. Math. (Basel), Volume 63 (1994) no. 3, pp. 211-215 | DOI | MR | Zbl

[30] Magaard, Kay; Malle, Gunter Irreducibility of alternating and symmetric squares, Manuscripta Math., Volume 95 (1998) no. 2, pp. 169-180 | DOI | MR | Zbl

[31] Magaard, Kay; Malle, Gunter; Tiep, Pham Huu Irreducibility of tensor squares, symmetric squares and alternating squares, Pacific J. Math., Volume 202 (2002) no. 2, pp. 379-427 | DOI | MR | Zbl

[32] Manivel, Laurent A note on certain Kronecker coefficients, Proc. Amer. Math. Soc., Volume 138 (2010) no. 1, pp. 1-7 | DOI | MR | Zbl

[33] Manivel, Laurent On rectangular Kronecker coefficients, J. Algebraic Combin., Volume 33 (2011) no. 1, pp. 153-162 | DOI | MR | Zbl

[34] Mészáros, Szabolcs; Wolosz, János Symmetric and exterior squares of hook representations, 2019 | arXiv

[35] Navarro, G. Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, 250, Cambridge University Press, Cambridge, 1998, x+287 pages | DOI

[36] Pak, Igor; Panova, Greta Bounds on certain classes of Kronecker and q-binomial coefficients, J. Combin. Theory Ser. A, Volume 147 (2017), pp. 1-17 | MR | Zbl

[37] Pak, Igor; Panova, Greta; Vallejo, Ernesto Kronecker products, characters, partitions, and the tensor square conjectures, Adv. Math., Volume 288 (2016), pp. 702-731 | MR | Zbl

[38] Regev, Amitai Lie superalgebras and some characters of S n , Israel J. Math., Volume 195 (2013) no. 1, pp. 31-35 | DOI | MR | Zbl

[39] Remmel, Jeffrey B. A formula for the Kronecker products of Schur functions of hook shapes, J. Algebra, Volume 120 (1989) no. 1, pp. 100-118 | DOI | MR | Zbl

[40] Remmel, Jeffrey B.; Whitehead, Tamsen On the Kronecker product of Schur functions of two row shapes, Bull. Belg. Math. Soc. Simon Stevin, Volume 1 (1994) no. 5, pp. 649-683 | MR | Zbl

[41] Ressayre, Nicolas Vanishing symmetric Kronecker coefficients, Beitr. Algebra Geom., Volume 61 (2020) no. 2, pp. 231-246 | DOI | MR | Zbl

[42] Rosas, Mercedes H. The Kronecker product of Schur functions indexed by two-row shapes or hook shapes, J. Algebraic Combin., Volume 14 (2001) no. 2, pp. 153-173 | DOI | MR | Zbl

[43] Saxl, Jan The complex characters of the symmetric groups that remain irreducible in subgroups, J. Algebra, Volume 111 (1987) no. 1, pp. 210-219 | DOI | MR | Zbl

[44] Serre, Jean-Pierre Linear representations of finite groups, Graduate Texts in Mathematics, 42, Springer-Verlag, New York-Heidelberg, 1977, x+170 pages | DOI | Numdam

[45] Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures (2018), p. viii+1208 (ICM 2018, August 1–9, 2018) | DOI

[46] Stanley, Richard P. Positivity problems and conjectures in algebraic combinatorics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295-319 | Zbl

[47] Taylor, Jay A note on skew characters of symmetric groups, Israel J. Math., Volume 221 (2017) no. 1, pp. 435-443 | DOI | MR | Zbl

[48] Thrall, R. M. On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math., Volume 64 (1942), pp. 371-388 | DOI | MR | Zbl

[49] Vallejo, Ernesto A diagrammatic approach to Kronecker squares, J. Combin. Theory Ser. A, Volume 127 (2014), pp. 243-285 | DOI | MR | Zbl

[50] Webb, Peter A course in finite group representation theory, Cambridge Studies in Advanced Mathematics, 161, Cambridge University Press, Cambridge, 2016, xi+325 pages | DOI | MR

[51] Wolosz, János Irreducible modules for symmetric groups that are summands of their exterior square, J. Algebra, Volume 518 (2019), pp. 304-320 | DOI | MR | Zbl

[52] Zisser, Ilan The character covering numbers of the alternating groups, J. Algebra, Volume 153 (1992) no. 2, pp. 357-372 | DOI | MR | Zbl

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