Trace cocharacters and the Kronecker products of Schur functions

✍ Scribed by J.O. Carbonara; L. Carini; J.B. Remmel

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 270 KB
• Volume: 260
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00013-9

✦ Synopsis

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M r (F ) of r × r matrices over a field F of characteristic zero are given by TC r,n = λ∈Λ r (n) χ λ ⊗ χ λ where χ λ ⊗ χ λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ r (n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ = (λ 1 • • • λ k ) with k r. We study the behavior of the sequence of trace cocharacters TC r,n .

In particular, we study the behavior of the coefficient of χ (ν,n-m) in TC r,n as a function of n where ν = (ν 1 • • • ν k ) is some fixed partition of m and nm ν k . Our main result shows that such coefficients always grow as a polynomial in n of degree r -1.