Consider the Sn-action on Rn given by permuting coordinates. This paper addresses the following problem: compute maxv,H|H∩Snv| as H⊂Rn ranges over all hyperplanes through the origin and v∈Rn ranges ...over all vectors with distinct coordinates that are not contained in the hyperplane ∑xi=0. We conjecture that for n≥3, the answer is (n−1)! for odd n, and n(n−2)! for even n. We prove that if p is the largest prime with p≤n, then maxv,H|H∩Snv|≤n!p. In particular, this proves the conjecture when n or n−1 is prime.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
For any field K and any transitive subgroup G of S8, let G act naturally on K(x1,…,x8) by permuting the variables. We prove that under some minor conditions K(x1,…,x8)G is always K-rational except G ...is A8 or G is isomorphic to PGL(2,7). We pay special attentions on the characteristic 2 cases.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We introduce a basis of the symmetric functions that evaluates to the (irreducible) characters of the symmetric group, just as the Schur functions evaluate to the irreducible characters of GLn ...modules. Our main result gives three different characterizations for this basis. One of the characterizations shows that the structure coefficients for the (outer) product of these functions are the stable Kronecker coefficients. The results in this paper focus on developing the fundamental properties of this basis.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Given an Eulerian digraph, we consider the genus distribution of its face-oriented embeddings. We prove that such distribution is log-concave for two families of Eulerian digraphs, thus giving a ...positive answer for these families to a question asked in Bonnington et al. (2002) 1. Our proof uses real-rooted polynomials and the representation theory of the symmetric group Sn. The result is also extended to some factorizations of the identity in Sn that are rotation systems of some families of one-face constellations.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Abstract
Brazil
et al
. ‘Maximal subgroups of infinite symmetric groups’,
Proc. Lond. Math. Soc. (3)
68
(1) (1994), 77–111 provided a new family of maximal subgroups of the symmetric group
$G(X)$
...defined on an infinite set
X
. It is easy to see that, in this case,
$G(X)$
contains subsemigroups that are not groups, but nothing is known about nongroup maximal subsemigroups of
$G(X)$
. We provide infinitely many examples of such semigroups.
This paper introduces new techniques for the efficient computation of discrete Fourier transforms (DFTs) of Sn−k-invariant functions on the symmetric group Sn. We uncover diamond- and leaf-rake-like ...structures in Young's seminormal and orthogonal representations leading to relatively expensive diamond and cheaper leaf-rake computations. These local computations constitute the basis of a reduction/induction process. We introduce a new anticipation technique that avoids diamond computations at the expense of only a small arithmetic overhead for leaf-rake computations. This results in local fast Fourier transforms (FFTs). Combining these local FFTs with a multiresolution scheme close related to the inductive version of Young's branching rule we obtain a global FFT algorithm that computes the DFT of Sn−k-invariant functions on Sn in linear time. More precisely, we show that for fixed k and all n≥2k DFTs of Sn−k-invariant functions on Sn can be computed in at most ck⋅Sn:Sn−k scalar multiplications and additions, where ck denotes a positive constant depending only on k. This run-time is order-optimal and improves Maslen's algorithm.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We establish a connection between two settings of representation stability for the symmetric groups Sn over C. One is the symmetric monoidal category Rep(S∞) of algebraic representations of the ...infinite symmetric group S∞=⋃nSn, related to the theory of FI-modules. The other is the family of rigid symmetric monoidal Deligne categories Rep_(St), t∈C, together with their abelian versions Rep_ab(St), constructed by Comes and Ostrik.
We show that for any t∈C the natural functor Rep(S∞)→Rep_ab(St) is an exact symmetric faithful monoidal functor, and compute its action on the simple representations of S∞. Considering the highest weight structure on Rep_ab(St), we show that the image of any object of Rep(S∞) has a filtration with standard objects in Rep_ab(St).
As a by-product of the proof, we give answers to the questions posed by P. Deligne concerning the cohomology of some complexes in the Deligne category Rep_(St), and their specializations at non-negative integers n.
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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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