Kronecker positivity and 2-modular representation theory

This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and other modular representation theoretic techniques on the study of Kronecker coefficients and hence verify Saxl’s conjecture for several large new families of partitions. In particular, we verify Saxl’s conjecture for all irreducible characters of S n \mathfrak {S}_n which are of 2-height zero.

[1]  Greta Panova,et al.  Kronecker products, characters, partitions, and the tensor square conjectures , 2013, 1304.0738.

[2]  Pham Huu Tiep,et al.  Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type , 2012, 1209.1768.

[3]  Alain Lascoux,et al.  Hecke algebras at roots of unity and crystal bases of quantum affine algebras , 1996 .

[4]  P. Shan,et al.  Categorifications and cyclotomic rational double affine Hecke algebras , 2013, 1305.4456.

[5]  Michela Varagnolo,et al.  On the decomposition matrices of the quantized Schur algebra , 1998 .

[6]  M. Fayers,et al.  The reducible Specht modules for the Hecke algebra $\mathcal{H}_{\mathbb{C},{-}1}(\mathfrak{S}_{n})$ , 2011, 1106.5602.

[7]  Christian Ikenmeyer.  The Saxl conjecture and the dominance order , 2015, Discret. Math..

[8]  Joan F. Tent,et al.  Alperin–McKay natural correspondences in solvable and symmetric groups for the prime $$p=2$$p=2 , 2017, 1707.02846.

[9]  B. Webster,et al.  ROUQUIER’S CONJECTURE AND DIAGRAMMATIC ALGEBRA , 2013, Forum of Mathematics, Sigma.

[10]  Matthias Christandl,et al.  The Spectra of Quantum States and the Kronecker Coefficients of the Symmetric Group , 2006 .

[11]  A diagrammatic approach to categorification of quantum groups II , 2009 .

[12]  M. Fayers,et al.  THE REDUCIBLE SPECHT MODULES FOR THE HECKE ALGEBRAHC,−1(Sn) , 2011 .

[13]  S. Lyle.  Some q-analogues of the Carter-Payne theorem , 2006, math/0604216.

[14]  G. James,et al.  The Irreducible Specht Modules in Characteristic 2 , 1999 .

[15]  M. Fayers.  Reducible Specht modules , 2004 .

[16]  A. Klyachko.  QUANTUM MARGINAL PROBLEM AND REPRESENTATIONS OF THE SYMMETRIC GROUP , 2004, quant-ph/0409113.

[17]  Jorn B. Olson.  McKay numbers and heights of characters. , 1976 .

[18]  Gordon James,et al.  A q‐Analogue of the Jantzen–Schaper Theorem , 1997 .

[19]  T. Okuyama,et al.  The heights of irreducible Brauer characters in 2-blocks of the symmetric groups , 2012 .

[20]  Chris D. Godsil,et al.  ALGEBRAIC COMBINATORICS , 2013 .

[21]  I. Losev.  Proof of Varagnolo–Vasserot conjecture on cyclotomic categories O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mat , 2013, Selecta Mathematica.

[22]  Victor Ginzburg,et al.  On the category 𝒪 for rational Cherednik algebras , 2002 .

[23]  J. Brundan,et al.  Graded decomposition numbers for cyclotomic Hecke algebras , 2009, 0901.4450.

[24]  R. Rouquier.  2-Kac-Moody algebras , 2008, 0812.5023.

[25]  I. Losev.  Proof of Varagnolo–Vasserot conjecture on cyclotomic categories $${\mathcal {O}}$$O , 2016 .

[26]  M. Fayers,et al.  Some reducible Specht modules for Iwahori–Hecke algebras of type A with q=−1 , 2008, 0806.1774.

[27]  Wolfgang Soergel,et al.  Koszul Duality Patterns in Representation Theory , 1996 .

[28]  Kai Meng Tan,et al.  A v-analogue of Peel's theorem☆ , 2004 .

[29]  Mark Sellke,et al.  The Saxl conjecture for fourth powers via the semigroup property , 2015, 1511.02387.

[30]  M. Fayers.  Irreducible Specht modules for Hecke algebras of type A , 2005 .

[31]  Andrew Mathas,et al.  Hecke algebras and Schur algebras of the symmetric group , 1999 .

[32]  M. Fayers.  An algorithm for semistandardising homomorphisms , 2011, 1109.4522.

[33]  Gordon James,et al.  The Representation Theory of the Symmetric Groups , 1977 .

[34]  Decompositions of some Specht modules I , 2018, 1810.10275.

[35]  G. James,et al.  Hecke Algebras of TypeAwithq=−1 , 1996 .

[36]  Weiqiang Wang,et al.  Graded Specht modules , 2009, 0901.0218.

[37]  Jørn B. Olsson,et al.  Combinatorics and representations of finite groups , 1993 .

[38]  J. Brundan,et al.  Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras , 2008, 0808.2032.

[39]  G. James,et al.  The Representation Theory of the Symmetric Group , 2009 .

[40]  Gordon James,et al.  Representations of Hecke Algebras of General Linear Groups , 1986 .

[41]  M. Fayers.  q-Analogues of regularisation theorems for linear and projective representations of the symmetric group , 2007 .

[42]  G. Murphy.  On decomposability of some Specht modules for symmetric groups , 1980 .

[43]  C. Bessenrodt.  Critical classes, Kronecker products of spin characters, and the Saxl conjecture , 2017, 1704.00707.

[44]  C. Bessenrodt,et al.  Multiplicity-free Kronecker products of characters of the symmetric groups , 2016, 1609.03596.

[45]  M. Fayers,et al.  Generalised column removal for graded homomorphisms between Specht modules , 2014, 1404.4415.

[46]  G. James,et al.  $q$-tensor space and $q$-Weyl modules , 1991 .

[47]  Greta Panova,et al.  Bounds on certain classes of Kronecker and q-binomial coefficients , 2017, J. Comb. Theory, Ser. A.

[48]  A. Kleshchev,et al.  An Interpretation of the Lascoux–Leclerc–Thibon Algorithm and Graded Representation Theory , 2009, 0910.5940.

[49]  Gordon James,et al.  Rouquier blocks , 2005 .

[50]  Aram W. Harrow,et al.  Nonzero Kronecker Coefficients and What They Tell us about Spectra , 2007 .

[51]  M. Christandl,et al.  Recoupling Coefficients and Quantum Entropies , 2012, 1210.0463.

[52]  Some new decomposable Specht modules , 2011, 1110.0296.

[53]  Michael Walter,et al.  Computing Multiplicities of Lie Group Representations , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[54]  M. Fayers.  On the irreducible Specht modules for Iwahori--Hecke algebras of type A with $q=-1$ , 2009, 0911.0342.

[55]  Victor Ginzburg,et al.  Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism , 2000 .

[56]  Raphael Rouquier.  q-Schur algebras and complex reflection groups, I , 2005 .