# Can the Jacobi-Trudi identity be understood as a BGG resolution?

The thought process that led me to this question is that the identity $$\left(\prod_i \frac1{1-x_i}\right)\left(\prod_i {1-x_i}\right)=1$$ can be understood as expressing exactness of the Koszul complex. This identity is rewritten by taking $\left(\prod_i \frac1{1-x_i}\right)$ as the generating function for the complete symmetric functions $h_n$ and $\left(\prod_i {1+x_i}\right)$ as the generating function for the elementary symmetric functions $e_n$.

Next we have the Jacobi-Trudi identity which expresses a Schur function as the determinant of a matrix whose entries are complete (or elementary) symmetric functions. Also the Specht module is sometimes constructed as a quotient (or submodule) of the trivial representation of the Young subgroup induced to a representation. This suggests that this is the start of a BGG resolution.

I imagine that if this works then it is well-known. Could I have some references? and where does line of thought lead?

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I already voted up Jim's answer and don't have one to add. I did work out some explicit matrices some time ago, so get in touch if you want them. The matrix entries are hard to work out if you use the usual determinant form of the Jacobi-Trudi, but if you use the raising operator form and accept a non-minimal resolution they are all +/- 1. –  Alexander Woo Jun 5 '10 at 20:25
For Koszul and the identity see also mathoverflow.net/questions/98621/… –  Alexander Chervov Jun 18 '12 at 19:12

Look at the short paper MR902299 (89a:17012) 17B10 (20C30) Zelevinski˘ı, A.V. [Zelevinsky, Andrei] (2-AOS-CY), Resolutions, dual pairs and character formulas. (Russian) Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 74–75, as well as the independent work by Kaan Akin (a former student of David Buchsbaum) including MR1194310 (94e:20059) 20G05 Akin, Kaan (1-OK), On complexes relating the Jacobi-Trudi identity with the Bernstein-Gel0fand-Gel0fand resolution. II. J. Algebra 152 (1992), no. 2, 417–426. A further refinement is given in MR1379204 (97b:20066) 20G05 Maliakas, Mihalis (1-AR), Resolutions and parabolic Schur algebras. J. Algebra 180 (1996), no. 3, 679–690.

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Thanks. I half thought I might see 'straightening' being used. –  Bruce Westbury Jun 6 '10 at 15:49