The restriction problem and the Frobenius transform

Restriction coefficients are the coefficients that appear when decomposing an irreducible GL_n-module into irreducible S_n-modules. The problem of finding a combinatorial formula for restriction coefficients is called the restriction problem and it is a central open problem in algebraic combinatorics. By suitably decategorifying Joyal’s analytic functor construction, we define a linear map called the Frobenius transform on the ring of symmetric functions. We then use the Frobenius transform to prove some vanishing results about restriction coefficients as well as an identity that relates restriction coefficients, Kronecker coefficients, and Littlewood-Richardson coefficients.

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