Permutation resolutions for Specht modules
Robert Boltje
and Robert Hartmann
DOI: 10.1007/s10801-010-0265-1
Abstract
For every composition λ of a positive integer r, we construct a finite chain complex whose terms are direct sums of permutation modules M μ for the symmetric group \mathfrak S r \mathfrak{S}_{r} with Young subgroup stabilizers \mathfrak S m \mathfrak{S}_{μ}. The construction is combinatorial and can be carried out over every commutative base ring k. We conjecture that for every partition λ the chain complex has homology concentrated in one degree (at the end of the complex) and that it is isomorphic to the dual of the Specht module S λ . We prove the exactness in special cases.
Pages: 141–162
Keywords: keywords symmetric group; permutation module; Specht module; resolution
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References
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2. Akin, K., Buchsbaum, D.A.: Characteristic-free representation theory of the general linear group. Adv. Math. 58, 149-200 (1985)
3. Akin, K., Buchsbaum, D.A.: Characteristic-free representation theory of the general linear group, II: homological considerations. Adv. Math. 72, 171-210 (1988)
4. Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Differential operators on the base affine space and a study of g-modules. In: Gelfand, I.M. (ed.) Lie Groups and Their Representations, pp. 21-64. Wiley, New York (1975)
5. Curtis, C.W., Reiner, I.: Methods of Representation Theory, vol.
1. Wiley, New York (1981)
6. Donkin, S.: Finite resolutions of modules for reductive algebraic groups. J. Algebra 101, 473-488 (1986)
7. Doty, S.R.: Resolutions of B modules. Indag. Math. 5(3), 267-283 (1994) J Algebr Comb (2011) 34: 141-162
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