Let X be a character table of the symmetric group Sn. It is shown that unless n = 4 or n = 6, there is a unique way to assign partitions of n to the rows and columns of X so that for all λ and ν, Xλν is equal to χ(ν), the value of the irreducible character of Sn labelled by λ on elements of cycle type ν. Analogous results are proved for alternating groups, and for the Brauer character tables of...

While we were graduate students, Marty Isaacs and I worked together on the character theory of finite groups, studying in particular the character degrees of finite p-groups. Somewhat later, my interests turned to ring theory and infinite group theory. On the other hand, Marty continued with character theory and soon became a leader in the field. Indeed, he has had a superb career as a research...

A character identity which relates irreducible values of the hyperoctahedral group \(B_n\) to those symmetric \(S_{2n}\) was recently proved by Lübeck and Prasad. Their proof is algebraic involves Lie theory. We present a short combinatorial this identity, as well generalization other wreath products.Mathematics Subject Classifications: 20C30, 20E22, 05E10Keywords: Character product, partition,...

For a Weyl group W , we investigate simple closed formulas (valid on elliptic conjugacy classes) for the character of the representation of W in the homology of a Springer fiber. We also give a formula (valid again on elliptic conjugacy classes) of the W -character of an irreducible discrete series representation with real central character of a graded affine Hecke algebra with arbitrary parame...

Let π be an irreducible representation of a real reductive group G. The global character θπ of π may be considered as a function on the regular semisimple elements of G. The global character determines π, and it is of great interest to compute it. For example Harish-Chandra found the discrete series representations of G by computing their characters. Fix a Cartan subgroup H of G and let D be th...

For a finite group G, the character degree graph ∆(G) is the graph whose vertices are the primes dividing the degrees of the ordinary irreducible characters of G, with distinct primes p and q joined by an edge if pq divides some character degree of G. We determine all graphs with four vertices that occur as ∆(G) for some nonsolvable group G. Along with previously known results on character degr...

Using vertex operators, we construct explicitly Lusztig’s Z[q, q−1]-lattice for the level one irreducible representations of quantum affine algebras of ADE type. We then realize the level one irreducible modules at roots of unity and show that the character is given by the Weyl-Kac character formula. 0. Introduction In [L1] and [L3] G. Lusztig proved that a quantum Kac-Moody algebra U defined o...

Frobenius began the study of representation theory and character theory of the symmetric groups Sn at the turn of the century [4]. In this study, He showed that there is one irreducible representation πλ of Sn corresponding to each partition λ of n and this correspondence gives all of irreducible representations of Sn up to equivalency. After this work, he also began the study of representation...

Let G be a compact, semi-simple Lie group and H a maximal rank reductive subgroup. The irreducible representations of G can be constructed as spaces of harmonic spinors with respect to a Dirac operator on the homogeneous space G/H twisted by bundles associated to the irreducible, possibly projective, representations of H. Here, we give a quick proof of this result, computing the index and kerne...