In this paper, we study relationships between the normalized characters of symmetric groups and the Boolean cumulants of Young diagrams. Specifically, we show that each normalized character is a ...polynomial of twisted Boolean cumulants with coefficients being non-negative integers, and conversely, that, when we expand a Boolean cumulant in terms of normalized characters, the coefficients are again non-negative integers. The main tool is Khovanov's Heisenberg category and the recently established connection of its center to the ring of functions on Young diagrams, which enables one to apply graphical manipulations to the computation of functions on Young diagrams. Therefore, this paper is an attempt to deepen the connection between the asymptotic representation theory and graphical categorification.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Permutation invariant Gaussian two-matrix models Barnes, George; Padellaro, Adrian; Ramgoolam, Sanjaye
Journal of physics. A, Mathematical and theoretical,
04/2022, Volume:
55, Issue:
14
Journal Article
Peer reviewed
Open access
Abstract
We construct the general permutation invariant Gaussian two-matrix model for matrices of arbitrary size
D
. The parameters of the model are given in terms of variables defined using the ...representation theory of the symmetric group
S
D
. A correspondence is established between the permutation invariant polynomial functions of the matrix variables (the observables of the model) and directed colored graphs, which sheds light on stability properties in the large
D
counting of these invariants. A refined counting of the graphs is given in terms of double cosets involving permutation groups defined by the local structure of the graphs. Linear and quadratic observables are transformed to an
S
D
representation theoretic basis and are used to define the convergent Gaussian measure. The perturbative rules for the computation of expectation values of graph-basis observables of any degree are given in terms of the representation theoretic parameters. Explicit results for a number of observables of degree up to four are given along with a Sage programme that computes general expectation values.
In this paper, we introduce higher symmetric simplicial complexity SCnΣ(K) of a simplicial complex K and higher symmetric combinatorial complexity CCnΣ(P) of a finite poset P. These are simplicial ...and combinatorial approaches to symmetric motion planning of Basabe - González - Rudyak - Tamaki. We prove that the symmetric simplicial complexity SCnΣ(K) is equal to symmetric topological complexity TCnΣ(|K|) of the geometric realization of K and the symmetric combinatorial complexity CCnΣ(P) is equal to symmetric topological complexity TCnΣ(|K(P)|) of the geometric realization of the order complex of P.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Given two positive integers n≥3 and t≤n, the permutations σ,π∈Sym(n) are t-setwise intersecting if they agree (setwise) on a t-subset of {1,2,…,n}. A family F⊂Sym(n) is t-setwise intersecting if any ...two permutations of F are t-setwise intersecting. Ellis (2012) 6 conjectured that if t≤n and F⊂Sym(n) is a t-setwise intersecting family, then |F|≤t!(n−t)! and equality holds only if F is a coset of a setwise stabilizer of a t-subset of {1,2,…,n}.
In this paper, we prove that if n≥11 and F⊂Sym(n) is 3-setwise intersecting, then |F|≤6(n−3)!. Moreover, we prove that the characteristic vector of a 3-setwise intersecting family of maximum size lies in the sum of the eigenspaces induced by the permutation module of Sym(n) acting on the 3-subsets of {1,2,…,n}.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Abstract
Motivated by questions originating from the study of a class of shallow student-teacher neural networks, methods are developed for the analysis of spurious minima in classes of gradient ...equivariant dynamics related to neural networks. In the symmetric case, methods depend on the generic equivariant bifurcation theory of irreducible representations of the symmetric group on
k
symbols,
S
k
; in particular, the standard representation of
S
k
. It is shown that spurious minima (non-global local minima) do not arise from spontaneous symmetry breaking but rather through a complex deformation of the landscape geometry that can be encoded by a generic
S
k
-equivariant bifurcation. We describe minimal models for forced symmetry breaking that give a lower bound on the dynamic complexity involved in the creation of spurious minima when there is no symmetry. Results on generic bifurcation when there are quadratic equivariants are also proved; this work extends and clarifies results of Ihrig & Golubitsky and Chossat, Lauterbach & Melbourne on the instability of solutions when there are quadratic equivariants.
The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and ...Peter Symonds defined a new invariant γG(M) for a finite dimensional module M of a finite group G which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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