Lecture 6 : Kronecker Product of Schur Functions – Part I
نویسنده
چکیده
The irreducible representations of Sn, i.e. the Specht modules are indexed by partitions λ of n. For any two partitions λ, μ of n, Sλ ⊗ Sμ = gλμνSν , for suitable integers gλμν . The actual values of these coefficients still eludes us. We look at a formula (admittedly messy), which gives the exact values of gλμν for simple shapes λ, μ.
منابع مشابه
The stability of the Kronecker product of Schur functions
In the late 1930’s Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n large enough, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n when ...
متن کاملThe Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes
The Kronecker product of two Schur functions sμ and sν , denoted by sμ ∗sν, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions μ and ν. The coefficient of sλ in this product is denoted by γ λ μν , and corresponds to the multiplicity of the irreducible character χ in χχ . We use Sergeev’s Formula for a S...
متن کاملOn the Kronecker Product of Schur Functions of Two Row Shapes
The Kronecker product of two homogeneous symmetric polynomials P1 and P2 is defined by means of the Frobenius map by the formula P1 ⊗ P2 = F (F−1P1)(F−1P2). When P1 and P2 are Schur functions sλ and sμ respectively, then the resulting product sλ ⊗ sμ is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the diagrams λ an...
متن کاملKronecker Coefficients for Some Near-rectangular Partitions
We give formulae for computing Kronecker coefficients occurring in the expansion of sμ ∗ sν , where both μ and ν are nearly rectangular, and have smallest parts equal to either 1 or 2. In particular, we study s(n,n−1,1) ∗ s(n,n), s(n−1,n−1,1) ∗ s(n,n−1), s(n−1,n−1,2) ∗ s(n,n), s(n−1,n−1,1,1) ∗ s(n,n) and s(n,n,1) ∗ s(n,n,1). Our approach relies on the interplay between manipulation of symmetric...
متن کاملKronecker Product Identities from D-finite Symmetric Functions
Using an algorithm for computing the symmetric function Kronecker product of D-finite symmetric functions we find some new Kronecker product identities. The identities give closed form formulas for trace-like values of the Kronecker product. Introduction In the process of showing how the scalar product of symmetric functions can be used for enumeration purposes, Gessel [3], proved that this pro...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2003