Celebratio Mathematica

G. Benkart and C. Lee: “Stability in modules for general linear Lie superalgebras,” Nova J. Algebra Geom. 2 : 4 (1993), pp. 383–409. MR 1285098 Zbl 0873.17003 article

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, C. L. Shader, and A. Ram: “Tensor product representations for orthosymplectic Lie superalgebras,” J. Pure Appl. Algebra 130 : 1 (August 1998), pp. 1–48. MR 1632811 Zbl 0932.17008 ArXiv math/9607232 article

We derive a general result about commuting actions on certain objects in braided rigid monoidal categories. This enables us to define an action of the Brauer algebra on the tensor space \( V^{\otimes k} \) which commutes with the action of the orthosymplectic Lie superalgebra \( \operatorname{spo}(V) \) and the orthosymplectic Lie color algebra \( \operatorname{spo}(V,\beta) \). We use the Brauer algebra action to compute maximal vectors in \( V^{\otimes k} \) and to decompose \( V^{\otimes k} \) into a direct sum of submodules \( T^{\lambda} \). We compute the characters of the modules \( T^{\lambda} \), give a combinatorial description of these characters in terms of tableaux, and model the decomposition of \( V^{\otimes k} \) into the submodules \( T^{\lambda} \) with a Robinson–Schensted–Knuth-type insertion scheme.

Arun Ram	Related
Chanyoung Lee Shader	Related

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Georgia Benkart

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