Representations of symmetric groups with non-trivial determinant

Ayyer, Arvind; Prasad, Amritanshu; Spallone, Steven

doi:10.1016/j.jcta.2017.03.004

Cited by 5 publications

(11 citation statements)

References 4 publications

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“…The most striking result of Ref. [2] is a closed formula for the number of chiral partitions of n. This answers an open question posed in [18,Exercise 7.55(b)]. In Section 3, we review this result as well as analogues of chiral partitions for all Coxeter groups.…”

Section: Summary Of Resultsmentioning

confidence: 68%

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Macdonald trees and determinants of representations for finite Coxeter groups

Ayyer¹,

Prasad²,

Spallone³

2018

Preprint

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Every irreducible odd dimensional representation of the n'th symmetric or hyperoctahedral group, when restricted to the (n−1)'th, has a unique irreducible odd-dimensional constituent. Furthermore, the subgraph induced by odd-dimensional representations in the Bratteli diagram of symmetric and hyperoctahedral groups is a binary tree with a simple recursive description. We survey the description of this tree, known as the Macdonald tree, for symmetric groups, from our earlier work. We describe analogous results for hyperoctahedral groups.A partition λ of n is said to be chiral if the corresponding irreducible representation V λ of Sn has non-trivial determinant. We review our previous results on the structure and enumeration of chiral partitions, and subsequent extension to all Coxeter groups by Ghosh and Spallone. Finally we show that the numbers of odd and chiral partitions track each other closely.

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Section: Summary Of Resultsmentioning

confidence: 68%

“…This article is mainly a survey of the main results of three papers [1] and [2] and [7] concerning the application of the dyadic arithmetic of partitions to the representation theory of finite Coxeter groups.…”

Section: Summary Of Resultsmentioning

confidence: 99%

Macdonald trees and determinants of representations for finite Coxeter groups

Ayyer¹,

Prasad²,

Spallone³

2018

Preprint

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“…In Section 3, for completeness, we indicate the solution to the determinant problem for dihedral groups. Section 4 reviews the work of [Mac71] and [APS17] for S n , and develops it further for application to the hyperoctahedral case.…”

Section: Moreover We Provementioning

confidence: 99%

“…The tools of this paper, and [APS17], have their genesis in the paper [Mac71] of Macdonald, who developed the arithmetic of partitions to give a closed formula for the number A(n) of odd-dimensional Specht modules (irreducible representations) of S n . Briefly, if n is expressed in binary as a sum of powers of 2, then A(n) is the product of those powers of 2.…”

Section: Introductionmentioning

confidence: 99%

“…This formula comes from characterizing the 2-core tower of "odd" partitions; we review this notion in Section 5.1. In [APS17], the authors describe a simple way to read off the determinant of a representation of S n from its 2-core tower, and in particular give a closed formula for the number B(n) of Specht modules whose determinant is the sign character. Since the number p(n) of partitions of n itself does not have a nice closed formula, it is remarkable that such formulas for A(n) and B(n) exist.…”

Section: Introductionmentioning

confidence: 99%

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Determinants of representations of Coxeter groups

Ghosh¹,

Spallone

2018

J Algebr Comb

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In [APS17], the authors characterize the partitions of n whose corresponding representations of S n have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups W . Namely, given a nontrivial multiplicative character ω of W , we give a closed formula for the number of irreducible representations of W with determinant ω. For Coxeter groups of type B n and D n , this is accomplished by characterizing the bipartitions associated to such representations.

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Spinorial representations of symmetric groups

Ganguly

Spallone

2020

Journal of Algebra

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Let G be a real compact Lie group, such that G = G 0 ⋊ C 2 , with G 0 simple. Here G 0 is the connected component of G containing the identity and C 2 is the cyclic group of order 2. We give a criterion whether an orthogonal representation π : G → O(V ) lifts to Pin(V ) in terms of the highest weights of π. We also calculate the first and second Stiefel-Whitney classes of the representations of the Orthogonal groups.

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Representations of symmetric groups with non-trivial determinant

Cited by 5 publications

References 4 publications

Macdonald trees and determinants of representations for finite Coxeter groups

Macdonald trees and determinants of representations for finite Coxeter groups

Determinants of representations of Coxeter groups

Spinorial representations of symmetric groups

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