DISTANCES IN GRAPHS OF PERMUTATIONS Dougherty, Steven T.; Gianello, Mia
The Rocky Mountain journal of mathematics,
4/2024, Volume:
54, Issue:
2
Journal Article
We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid Inc(N) of ...strictly increasing functions. Our first main result states that these series are rational functions in two variables. A key is to introduce also suitable submonoids of Inc(N) and to compare invariant filtrations induced by their actions. Extending a result by Hillar and Sullivant, we show that any ideal that is invariant under these submonoids admits a Gröbner basis consisting of finitely many orbits. As our second main result we prove that the Krull dimension and multiplicity of ideals in an invariant filtration grow eventually linearly and exponentially, respectively, and we determine the terms that dominate this growth.
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We construct an explicit isomorphism between (truncations of) quiver Hecke algebras and Elias–Williamson's diagrammatic endomorphism algebras of Bott–Samelson bimodules. As a corollary, we deduce ...that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan–Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. We hence give an elementary and more explicit proof of the main theorem of Riche–Williamson's recent monograph and extend their categorical equivalence to cyclotomic quiver Hecke algebras, thus solving Libedinsky–Plaza's categorical blob conjecture.
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Let G be the symmetric group of degree n. Let ω(G) be the maximal size of a subset S of G such that 〈x,y〉=G whenever x,y∈S and x≠y and let σ(G) be the minimal size of a family of proper subgroups of ...G whose union is G. We prove that both functions σ(G) and ω(G) are asymptotically equal to 12(nn/2) when n is even. This, together with a result of S. Blackburn, implies that σ(G)/ω(G) tends to 1 as n→∞. Moreover, we give a lower bound of n/5 on ω(G) which is independent of the classification of finite simple groups. We also calculate, for large enough n, the clique number of the graph defined as follows: the vertices are the elements of G and two vertices x,y are connected by an edge if 〈x,y〉≥An.
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We use the combinatorial way to give an explicit expression for the product of the class of cycles of length three with an arbitrary class of cycles. In addition, an explicit formula for the ...coefficient of an arbitrary class in the expansion of the product of an arbitrary class by the class of cycles of length three is given.
We give new bounds and asymptotic estimates for Kronecker and Littlewood–Richardson coefficients. Notably, we resolve Stanley's questions on the shape of partitions attaining the largest Kronecker ...and Littlewood–Richardson coefficients. We apply the results to asymptotics of the number of standard Young tableaux of skew shapes.
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The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. In this paper, we get a formula for the signed enumeration of alternating descents ...and in our proof we need a signed convolution type identity involving the Eulerian polynomials. When n is even, we give a more general multivariate version and we also get a formula for the signed enumeration of the alternating major index. We generalize our results to the case when alternating descents are summed up with sign over the elements in classical Weyl groups.
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38.
A simplified presentation of Specht modules Brauner, Sarah; Friedmann, Tamar
Journal of pure and applied algebra,
July 2022, 2022-07-00, Volume:
226, Issue:
7
Journal Article
Peer reviewed
Open access
Fulton and Kraskiewicz gave a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. We simplify this presentation by showing that it can be generated ...by a single relation for each pair of columns of a tableau with ordered columns, thereby significantly reducing the number of generators given in the original construction. Our presentation applies to all Specht modules, and is of a similar nature to a recent result by Friedmann-Hanlon-Stanley-Wachs that applies to staircase partitions. We show that our presentation implies the Friedmann-Hanlon-Stanley-Wachs presentation.
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On triangles in derangement graphs Meagher, Karen; Razafimahatratra, Andriaherimanana Sarobidy; Spiga, Pablo
Journal of combinatorial theory. Series A,
20/May , Volume:
180
Journal Article
Peer reviewed
Open access
Given a permutation group G, the derangement graph ΓG of G is the Cayley graph with connection set the set of all derangements of G. We prove that, when G is transitive of degree at least 3, ΓG ...contains a triangle.
The motivation for this work is the question of how large can be the ratio of the independence number of ΓG to the size of the stabilizer of a point in G. We give examples of transitive groups where this ratio is maximum.
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This paper provides a complete determination of which of the alternating groups An and the symmetric groups Sn occur as the automorphism group of some regular or chiral map on an orientable surface, ...and which of them occur as the automorphism group of a regular map on a non-orientable surface. The situation for some given types (m,k) is also considered, where k is the valency and m is the face-size, with special focus on types with m=3, and more particularly with (m,k)=(3,7) or (3,8), or their duals. Some observations are made also about what happens for regular and orientably-regular maps with given valency, and for regular and chiral polyhedra. Much but certainly not all of what is presented follows from theorems in previous papers by the author and others, and this one brings them and some new observations together into a single reference.
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