On Kronecker products of spin characters of the double covers of the symmetric groups

Abstract

In this article, restrictions on the constituents of Kronecker products of spin characters of the double covers of the symmetric groups are derived. This is then used to classify homogeneous and irreducible products of spin characters; as an application of this, certain homogeneous 2-modular tensor products for the symmetric groups are described.

On mixed products of complex characters of the double covers of the symmetric groups

“…We call a character of S n "homogeneous" if it is of the form c λ resp. "almost homogeneous" if it is of the form c λ for some λ ∈ D(n) and c ∈ N. We will need the following combinatorial result (see [2], Lemma 4.1 and its proof).…”

“…Recently, in [2] some results have been obtained on products of spin characters of S n which led to a classification of homogeneous spin products. Here, homogeneous products do occur for all triangular numbers n, but non-trivial irreducible products occur only for n = 6.…”

Critical classes, Kronecker products of spin characters, and the Saxl conjecture

“…(v) While the situations above are the ones used in the later sections, it should be noted that there many more such instances, and we mention just a few examples. For G = GL (3,2), the pair of classes of elements of order 7 is detecting for the pair of characters of degree 3. For G = PSL (2,11), the pair of classes of elements of order 5 (order 11, resp.)…”

“…An important link is made as a consequence of an identity found in an earlier investigation of homogeneous Kronecker products. While in [1] it was shown that there are no nontrivial homogeneous products of S n -characters, for the double covers S n of the symmetric groups nontrivial homogeneous spin products do occur for all triangular numbers n [2]. In [3], also nontrivial homogeneous mixed products of complex characters for the double covers S n were found, i.e., products of a non-faithful character of S n with a spin character.…”

On the Durfee size of Kronecker products of characters of the symmetric group and its double covers

“…Products with the basic spin character have been determined in [18]. A rectangular hull result for spin products was proved in [3], and this was then applied to classify the homogeneous spin products. Also, irreducible mixed products have been determined in [4].…”