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Powers of the Vandermonde determinant, Schur functions, and the dimension game

Abstract : Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give a recursive approach for computing the coefficient of the Schur function $s_μ$ in the decomposition of an even power of the Vandermonde determinant in $n+1$ variables in terms of the coefficient of the Schur function $s_λ$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $μ$ is obtained from the Young diagram of $λ$ by adding a tetris type shape to the top or to the left.
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https://hal.inria.fr/hal-01215094
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Cristina Ballantine. Powers of the Vandermonde determinant, Schur functions, and the dimension game. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.87-98, ⟨10.46298/dmtcs.2893⟩. ⟨hal-01215094⟩

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