The irreducible tensor representations of \(\text{gl}(m| 1)\) and their generic homology. (English) Zbl 1041.17005
From the introduction: In classical representation theory, all irreducible finite-dimensional representations of the Lie algebras \(\text{gl}(n)\) can be constructed using Young symmetrizers as operators on tensor powers of the standard representation of \(\text{gl} (n)\) and twisting by powers of the determinant character. For the Lie superalgebras \(\text{gl} (m| n)\), these operations are not sufficient to construct all irreducible finite-dimensional representations. In this note we describe how a larger family of tensor operations can be used to construct all irreducible finite-dimensional representations of \(\text{gl} (m| 1)\), and then use our constructions to calculate the homology of these and other basic representations under the generic actions of \(\text{gl}_1 (m| 1)\) and \(\text{gl}_{-1} (m| 1)\). This approach, developed in [K. Akin and J. Weyman, J. Algebra 197, 559–583 (1997; Zbl 0898.17003)] provides a curious link between the representation theory of \(\text{gl} (m| n)\) and natural tensor complexes used in construction of minimal resolutions associated to generic determinantal ideals.
The family of tensor operations we use consists of the symmetry operators realizing the Pieri maps, combined with trace and evaluation maps on mixed, covariant and contravariant tensors on the standard representation of \(\text{gl} (m| 1)\). From these basic operations, we construct complexes of tensor representations of \(\text{gl} (m| 1)\) which contain all finite-dimensional irreducible \(\text{gl} (m| 1)\)-modules as their cycles. For atypical weights, these complexes are double infinite exact complexes, and their right or left truncations give resolutions which realize over \(\text{gl} (m| 1)\) the character formulas of Bernstein and Leites. In Section 4, we exploit the relationship between our constructions and general tensor complexes built from Schur complexes to compute generic homology, over the coordinate algebra of affine \(m\)-space, of the irreducible modules and Kac modules of \(\text{gl} (m| 1)\).
The family of tensor operations we use consists of the symmetry operators realizing the Pieri maps, combined with trace and evaluation maps on mixed, covariant and contravariant tensors on the standard representation of \(\text{gl} (m| 1)\). From these basic operations, we construct complexes of tensor representations of \(\text{gl} (m| 1)\) which contain all finite-dimensional irreducible \(\text{gl} (m| 1)\)-modules as their cycles. For atypical weights, these complexes are double infinite exact complexes, and their right or left truncations give resolutions which realize over \(\text{gl} (m| 1)\) the character formulas of Bernstein and Leites. In Section 4, we exploit the relationship between our constructions and general tensor complexes built from Schur complexes to compute generic homology, over the coordinate algebra of affine \(m\)-space, of the irreducible modules and Kac modules of \(\text{gl} (m| 1)\).
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
Citations:
Zbl 0898.17003References:
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