Abstract: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.
“…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%
“…This is used to explain the results for all other Coxeter groups, which are taken from Ref. [7], and appear in later subsections. 3,5,4,8,12,20,8,16,24,40,32,64,88,152,16,32,48, .…”
Section: Determinants Of Representations Of Coxeter Groupsmentioning
confidence: 99%
“…The determinants of the representations of B n can be computed either through the Solomon formulas, combined with the Frobenius character formula, or through a well-known formula for the determinant of an induced representation. (See [7], for example.) Let…”
Section: Solomon Principlementioning
confidence: 99%
“…For n as in (1), let n = n − . The following equations compute N sgn 0 (n) in all cases [7]. Let k = v 2 (n ).…”
Section: Solomon Principlementioning
confidence: 99%
“…The dihedral case is facile, and the exceptional cases are finite in number and the determinants may be computed by available character tables. See [7] for these remaining cases. Moreover, a(n)/b(n + 2) = 1 if and only if n is divisible by 4.…”
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.
“…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%
“…This is used to explain the results for all other Coxeter groups, which are taken from Ref. [7], and appear in later subsections. 3,5,4,8,12,20,8,16,24,40,32,64,88,152,16,32,48, .…”
Section: Determinants Of Representations Of Coxeter Groupsmentioning
confidence: 99%
“…The determinants of the representations of B n can be computed either through the Solomon formulas, combined with the Frobenius character formula, or through a well-known formula for the determinant of an induced representation. (See [7], for example.) Let…”
Section: Solomon Principlementioning
confidence: 99%
“…For n as in (1), let n = n − . The following equations compute N sgn 0 (n) in all cases [7]. Let k = v 2 (n ).…”
Section: Solomon Principlementioning
confidence: 99%
“…The dihedral case is facile, and the exceptional cases are finite in number and the determinants may be computed by available character tables. See [7] for these remaining cases. Moreover, a(n)/b(n + 2) = 1 if and only if n is divisible by 4.…”
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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