Abstract
We provide a non-recursive, combinatorial classification of multiplicity-free skew Schur polynomials. These polynomials are $GL_n$, and $SL_n$, characters of the skew Schur modules. Our result extends work of H. Thomas--A. Yong, and C. Gutschwager, in which they classify the multiplicity-free skew Schur functions.
Highlights
The skew Schur polynomials are a fundamental family of symmetric polynomials whose connection to representation theory was first studied by I
This family is indexed by skew partitions λ/μ, where λ = (λ1 λ2 · · · λn > 0) and μ = (μ1 μ2 · · · μm > 0) are partitions with μ ⊆ λ
If neither (λ∗)∨ nor μ∗ is a rectangle of shortness 1, we have shown that sλ∗/μ∗ (x1, . . . , xρ(λ∗/μ∗)+1) is not multiplicity-free
Summary
The skew Schur polynomials are a fundamental family of symmetric polynomials whose connection to representation theory was first studied by I. This family is indexed by skew partitions λ/μ, where λ = (λ1 λ2 · · · λn > 0) and μ = (μ1 μ2 · · · μm > 0) are partitions with μ ⊆ λ (that is, m n and μi λi for 1 i m). The skew Schur polynomial of shape λ/μ is the generating function (1). Xn) we recover the Schur polynomials; the sλ(x1, . Ν where the sum is over partitions ν with (ν) n and the coefficients cλμ,ν are the celebrated Littlewood–Richardson coefficients. The expansion (2) is multiplicity-free if cλμ,ν ∈ {0, 1} for all ν with (ν) n. Our Theorem 1.11 gives a complete, non-recursive answer to this question.
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