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The Saxl Conjecture and the Dominance Order
[article]

2015
*
arXiv
*
pre-print

We make progress on this

arXiv:1410.6549v2
fatcat:cysgx2fslfg7jdcr2x4uagfg5i
*conjecture*by proving*the*occurrence of all those irreducibles which correspond to partitions that are comparable to*the*staircase partition in*the**dominance**order*. ... In 2012 Jan*Saxl**conjectured*that all irreducible representations of*the*symmetric group occur in*the*decomposition of*the*tensor square of*the*irreducible representation corresponding to*the*staircase ... We use induction on d, where*the*base case d = 1 is trivial. If ν has at most n columns, then*the*statement holds by Theorem 2.1 because ν is*dominated*by ̺(n). ...##
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A Tensor-Cube Version of the Saxl Conjecture
[article]

2022
*
arXiv
*
pre-print

*The*

*Saxl*

*conjecture*asserts that every irreducible representation S^λ of

*the*symmetric group S_N appears as a subrepresentation of

*the*tensor square S^ρ_n⊗ S^ρ_n. ... Let n be a positive integer,

*and*let ρ_n = (n, n-1, n-2, ..., 1) be

*the*"staircase" partition of size N = n+1 2. ...

*The*

*Saxl*

*conjecture*serves as one benchmark for results in this area. ...

##
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On the Saxl graph of a permutation group

2018
*
Mathematical proceedings of the Cambridge Philosophical Society (Print)
*

In this paper we introduce

doi:10.1017/s0305004118000610
fatcat:zraz3gzykjeurgnibsiktbytgi
*and*study an associated graph Σ(G), which we call*the**Saxl*graph of G.*The*vertices of Σ(G) are*the*points of Ω,*and*two vertices are adjacent if they form a base for G. ... For example,*the**conjecture*holds when G = Sn or An (with n > 12)*and**the*point stabiliser of G is a primitive subgroup. ... To see this, suppose γ(G) = k*and*let Λ be a total*dominating*set of size k. There are k · val(G)*ordered*pairs (α, β) such that α ∈ Λ*and*β is adjacent to α. ...##
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On the Saxl graph of a permutation group
[article]

2018
*
arXiv
*
pre-print

In this paper we introduce

arXiv:1712.03688v2
fatcat:eluqiqeyrngjzc5nwej5ml7b3q
*and*study an associated graph Σ(G), which we call*the**Saxl*graph of G.*The*vertices of Σ(G) are*the*points of Ω,*and*two vertices are adjacent if they form a base for G. ... For example,*the**conjecture*holds when G=S_n or A_n (with n>12)*and**the*point stabiliser of G is a primitive subgroup. ... To see this, suppose γ(G) = k*and*let Λ be a total*dominating*set of size k. There are k · val(G)*ordered*pairs (α, β) such that α ∈ Λ*and*β is adjacent to α. ...##
###
Critical classes, Kronecker products of spin characters, and the Saxl conjecture
[article]

2017
*
arXiv
*
pre-print

Via this tool, we make progress on

arXiv:1704.00707v1
fatcat:4qkisflgmbbndbkn7zgs44u2uu
*the**Saxl**conjecture*; this claims that for a triangular number n,*the*square of*the*irreducible character of*the*symmetric group S_n labelled by*the*staircase contains ... Highlighting*the*use of critical classes, we consider constituents in Kronecker products, in particular of spin characters of*the*double covers of*the*symmetric*and*alternating groups. ... staircase in*dominance**order*. ...##
###
Kronecker positivity and 2-modular representation theory

2021
*
Transactions of the American Mathematical Society. Series B
*

Secondly, we apply these results

doi:10.1090/btran/70
fatcat:emqr2ukvwnbohcof2pirn5mc2y
*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. ... We are also very grateful to*the*referees for their helpful comments*and*suggestions. ...##
###
Critical classes, Kronecker products of spin characters, and the Saxl conjecture

2018
*
Algebraic Combinatorics
*

Via this tool, we make progress on

doi:10.5802/alco.18
fatcat:46c2d3lrszehxfqetm34l6l7mm
*the**Saxl**conjecture*; this claims that for a triangular number n,*the*square of*the*irreducible character of*the*symmetric group Sn labelled by*the*staircase contains ... Using critical conjugacy classes, we find a new criterion for constituents in Kronecker products of spin characters of*the*double covers of*the*symmetric*and*alternating groups. ... Thanks go to John Stembridge for sharing his SF package, as well as to*the*referees for their suggestions*and*comments. ...##
###
Saxl Conjecture for triple hooks
[article]

2021
*
arXiv
*
pre-print

Firstly, we show that

arXiv:1811.10967v4
fatcat:5s7s2avnunde5dx2vei5h3l3ka
*the*probability that a partition is comparable in*dominance**order*to*the*staircase partition tends to zero as*the*staircase partition grows. ... We make some progresses on*Saxl**conjecture*. ... Acknowledgments We are very grateful to*the*editors*and*referees for their valuable comments*and*suggestions. ...##
###
Page 3666 of Mathematical Reviews Vol. , Issue 94g
[page]

1994
*
Mathematical Reviews
*

*The*argument used for

*the*latter classifica- tion relies on a theorem of N. F. J. Inglis, M. J. Liebeck

*and*

*Saxl*{Math. Z. 192 (1986), no. 3, 329-337; MR 87j:20022]. ... Oxford Ser. (2) 22 (1971), 551-557; MR 48 #5926],

*and*

*the*further reduction to

*the*classification of those with an almost simple or affine auto- morphism group by J.

*Saxl*, K. ...

##
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Kronecker products, characters, partitions, and the tensor square conjectures

2016
*
Advances in Mathematics
*

We study

doi:10.1016/j.aim.2015.11.002
fatcat:jzctpinsivbozj3yko7r4qhb7q
*the*remarkable*Saxl**conjecture*which states that tensor squares of certain irreducible representations of*the*symmetric groups Sn contain all irreducibles as their constituents. ... It is often referred as "classic",*and*"one of*the*last major open problems" in algebraic combinatorics [BWZ, Reg]. ... We are very grateful to Jan*Saxl*for telling us about his*conjecture*,*and*to Jonah Blasiak, Matt Boylan, Bob Guralnick, Stephen DeSalvo, Christian Ikenmeyer, Hari Narayanan, Rosa Orellana, Sinai Robins ...##
###
Kronecker positivity and 2-modular representation theory
[article]

2019
*
arXiv
*
pre-print

Secondly, we apply these results

arXiv:1903.07717v3
fatcat:icgy4tyql5ghjdxbi46n2gaiya
*and*other modular representation theoretic techniques on*the*study of Kronecker coefficients*and*hence verify Saxl's*conjecture*for a large new class of partitions. ... 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. ...*The*second author would like to thank both*the*Alexander von Humboldt Foundation*and**the*Leibniz Universität Hannover for financial support*and*an enjoyable summer. ...##
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A diagrammatic approach to Kronecker squares

2014
*
Journal of combinatorial theory. Series A
*

*The*first is a contribution to

*Saxl*

*conjecture*, which asserts that

*the*Kronecker square of

*the*staircase partition, contains every irreducible character of

*the*symmetric group as a component. ... In this paper we apply a method of Robinson

*and*Taulbee for computing Kronecker coefficients together with other ingredients

*and*show that

*the*multiplicity of each component in a Kronecker square can be ... Acknowledgements I am grateful to Christine Bessenrodt, Igor Pak

*and*Miguel Raggi for helpful suggestions. ...

##
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Finite simple groups and finite primitive permutation groups

1983
*
Bulletin of the Australian Mathematical Society
*

*The*classification of

*the*finite simple groups has had farreaching consequences for many branches of algebra. ... This paper is a discussion of several problems about primitive permutation groups which have been solved using

*the*simple group classification. ... These various bounds can be compared easily using

*the*following table, where (log J\n))* denotes

*the*

*order*of

*the*

*dominant*term in

*the*asymptotic expansion of log fin) . ...

##
###
A diagrammatic approach to Kronecker squares

2014
*
Discrete Mathematics & Theoretical Computer Science
*

(|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ for all $k$

doi:10.46298/dmtcs.2416
fatcat:b5j7rupmofdrxgv343glwgjx3m
*and*that there is an interval of*the*form $[c, \infty)$ in which $s_{\overline{ν}}$ is polynomial of degree $d$ with leading coefficient ... In this paper we improve a method of Robinson*and*Taulbee for computing Kronecker coefficients*and*show that for any partition $\overline{ν}$ of $d$ there is a polynomial $k_{\overline{ν}}$ with rational ... Acknowledgements I am grateful to Christine Bessenrodt, Igor Pak*and*Miguel Raggi for helpful suggestions. ...##
###
Resolution of singularities and modular Galois theory

2000
*
Bulletin of the American Mathematical Society
*

I shall also formulate several

doi:10.1090/s0273-0979-00-00892-2
fatcat:t2l4u2iyejggvhvg4jzx4qzyqi
*conjectures**and*suggest numerous thesis problems. ... I shall sketch a brief history of*the*desingularization problem from Riemann thru Zariski to Hironaka, including*the*part I played in it*and**the*work on Galois theory which this led me to,*and*how that ...*The*weaker versions of these two*Conjectures*may be stated as: Birational*Domination*Global*Conjecture*. ...
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