Celebratio Mathematica

G. Benkart and J. Stroomer: “Tableaux and insertion schemes for spinor representations of the orthogonal Lie algebra \( so(2r+1,\mathbb{C}) \),” J. Combin. Theory Ser. A 57 : 2 (July 1991), pp. 211–237. MR 1111558 Zbl 0747.17006 article

G. Benkart and J. Stroomer: “A combinatorial model for tensor products of the spin representation,” pp. 37–51 in Hadronic mechanics and nonpotential interactions (13–17 August 1990, Cedar Falls, IA), part 1: Mathematics. Edited by H. C. Myung. Nova Science Publishers, 1992. MR 1269550 Zbl 0813.17006 incollection

Jeffrey D. Stroomer	Related
Hyo Chul Myung	Related

G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Lee, and J. Stroomer: “Tensor product representations of general linear groups and their connections with Brauer algebras,” J. Algebra 166 : 3 (June 1994), pp. 529–567. For J. Marshall Osborn and Louis Solomon on their 60th birthdays. MR 1280591 Zbl 0815.20028 article

For the complex general linear group \( G = GL(r,\mathbb{C}) \) we investigate the tensor product module \[\textstyle T = (\bigotimes^p V)\otimes(\bigotimes^q V^*) \] of \( p \) copies of its natural representation \( V = \mathbb{C}^r \) and \( q \) copies of the dual space \( V^* \) of \( V \). We describe the maximal vectors of \( T \) and from that obtain an explicit decomposition of \( T \) into its irreducible \( G \)-summands. Knowledge of the maximal vectors allows us to determine the centralizer algebra \( \mathscr{C} \) of all transformations on \( T \) commuting with the action of \( G \), to construct the irreducible \( \mathscr{C} \)-representations, and to identify \( \mathscr{C} \) with a certain subalgebra \( \mathscr{B}^{(r)}_{p,q} \) of the Brauer algebra \( \mathscr{B}(r)_{p+q} \).

Jeffrey D. Stroomer	Related
Thomas Michael Halverson	Related
Louis Solomon	Related
Robert Edgar Leduc	Related
Chanyoung Lee Shader	Related
Manish Chakrabarti	Related
J. Marshall Osborn	Related

G. Benkart, F. Sottile, and J. Stroomer: “Tableau switching: Algorithms and applications,” J. Combin. Theory Ser. A 76 : 1 (October 1996), pp. 11–43. MR 1405988 Zbl 0858.05099 article

We define and characterize switching, an operation that takes two tableaux sharing a common border and “moves them through each other” giving another such pair. Several authors, including James and Kerber, Remmel, Haiman, and Shimozono, have defined switching operations; however, each of their operations is somewhat different from the rest and each imposes a particular order on the switches that can occur. Our goal is to study switching in a general context, thereby showing that the previously defined operations are actually special instances of a single algorithm. The key observation is that switches can be performed in virtually any order without affecting the final outcome. Many known proofs concerning the jeu de taquin, Schur functions, tableaux, characters of representations, branching rules, and the Littlewood–Richardson rule use essentially the same mechanism. Switching provides a common framework for interpreting these proofs. We relate Schützenberger’s evacuation procedure to switching and in the process obtain further results concerning evacuation. We define reversal, an operation which extends evacuation to tableaux of arbitrary skew shape, and apply reversal and related mappings to give combinatorial proofs of various symmetries of Littlewood–Richardson coefficients.

Jeffrey D. Stroomer	Related
Frank Sottile	Related

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Georgia Benkart

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