Multiplicity of trivial and sign representations of $S_n$ in hook-shaped representations of $GL_n$.

Abstract: Let $W_\lambda$ be an irreducible representation of $GL_n$ (for partition $\lambda$ with $\leq n$ parts). Let $V_\mu$ be an irreducible representation of $S_n$ (for partition $\mu \vdash n$). Then $$W_\lambda= \sum_{\mu \vdash n} r_{\lambda \mu} V_\mu.$$ The coefficients $r_{\lambda\mu}$ are the restriction coefficients. The restriction problem is to find combinatorial objects that these coefficients count. We find such objects when $\lambda$ is the hook shape and $\mu=(n)$ or $\mu= (1^n)$ using the theory of character polynomials and a simple sign-reversing involution.

combinatoricsrings and algebrasrepresentation theory

Audience: researchers in the topic

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ARCSIN - Algebra, Representations, Combinatorics and Symmetric functions in INdia

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