A combinatorial description of the quantum Désarménien matrix via the q-Schur algebra

The entries of the quantum Désarménien matrix, given in [A. Stokke, A quantum version of the Désarménien matrix, J. Algebraic Combin. 22 (2005) 303–316], are defined using elements in the quantum hyperalgebra. By exhibiting a map from the plus part of the quantum hyperalgebra to the q-Schur algebra, we describe the entries of this matrix using the q-Schur algebra. We then use the q-Schur algebra to give a combinatorial description of the entries in the matrix using Young tableaux.