Irreducible tensor products for symmetric groups in characteristic 2

Abstract

We consider non‐trivial irreducible tensor products of modular representations of a symmetric group Σn in characteristic 2 for even n completing the proof of a classification conjecture of Gow and Kleshchev about such products.

A note on vertices of indecomposable tensor products

“…Then by [2, Main Theorem] we have p = 2 and n is even. The exact tensor products which can arise in this way are described in [11], proving a conjecture of Gow and Kleshchev. It follows from that description, that then n = 2m for some odd integer m and that U can be chosen to be the basic spin module (labelled by the partition (m + 1, m − 1)).…”

Irreducible Tensor Products for Alternating Groups in Characteristic 5

“…The number of normal and conormal nodes of a partition are related by following result (see [25,Lemma 2.8]). Lemma 2.12.…”

“…If n is odd then D 1 ⊆ End F (D βn ) since in this case β n is not a JS-partition. If n is even then n ≡ 0 mod 4 by Lemma 2.2 and so D 1 ⊆ End F (D βn ) from [25,Lemma 7.1]. If λ has at least 4 normal nodes if n is odd or at least 5 normal nodes if n is even then D 3 1 ⊆ End F (D λ ).…”

“…From [18, Lemmas 4.4 and 6.1] it can be checked that ε 1−i (ẽ i λ) = 3 and ε i (ẽ 1−i λ) = x with 1 ≤ x ≤ 3. By [18,Lemma 4.8] it then follows that dim End S n−2 (D λ ) ≥ 5 + x.…”

“…Note that n ≥ 5 since λ ∈ P 2 (n) \ P A 2 (n) and it has two normal nodes. If M ∼ = 1|1 then M ∼ = 1↑ Sn An (for example from [18,Lemma 3.24]). So, since λ ∈ P A 2 (n), dim Hom Sn (M, End F (D λ )) = dim End An (E λ ) = 1.…”