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### The Saxl Conjecture and the Dominance Order [article]

Christian Ikenmeyer
2015 arXiv   pre-print
We make progress on this 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).  ...

### A Tensor-Cube Version of the Saxl Conjecture [article]

Nate Harman, Christopher Ryba
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.  ...

### On the Saxl graph of a permutation group

TIMOTHY C. BURNESS, MICHAEL GIUDICI
2018 Mathematical proceedings of the Cambridge Philosophical Society (Print)
In this paper we introduce 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 &gt; 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.  ...

### On the Saxl graph of a permutation group [article]

Timothy C. Burness, Michael Giudici
2018 arXiv   pre-print
In this paper we introduce 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]

Christine Bessenrodt
2017 arXiv   pre-print
Via this tool, we make progress on 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

C. Bessenrodt, C. Bowman, L. Sutton
2021 Transactions of the American Mathematical Society. Series B
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.  ...  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

Christine Bessenrodt
2018 Algebraic Combinatorics
Via this tool, we make progress on 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]

Xin Li
2021 arXiv   pre-print
Firstly, we show that 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.  ...

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

Igor Pak, Greta Panova, Ernesto Vallejo
2016 Advances in Mathematics
We study 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]

C. Bessenrodt, C. Bowman, L. Sutton
2019 arXiv   pre-print
Secondly, we apply these results 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.  ...

### A diagrammatic approach to Kronecker squares

Ernesto Vallejo
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.  ...

### Finite simple groups and finite primitive permutation groups

Cheryl E. Praeger
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

Ernesto Vallejo
2014 Discrete Mathematics & Theoretical Computer Science
(|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$for all$k$and that there is an interval of the form$[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

Shreeram S. Abhyankar
2000 Bulletin of the American Mathematical Society
I shall also formulate several 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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