Can the Jacobi-Trudi identity be understood as a BGG resolution? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:14:46Z http://mathoverflow.net/feeds/question/27176 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27176/can-the-jacobi-trudi-identity-be-understood-as-a-bgg-resolution Can the Jacobi-Trudi identity be understood as a BGG resolution? Bruce Westbury 2010-06-05T17:52:07Z 2010-06-09T12:36:08Z <p>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 $e_n$.</p> <p>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.</p> <p>I imagine that if this works then it is well-known. Could I have some references? and where does line of thought lead?</p> http://mathoverflow.net/questions/27176/can-the-jacobi-trudi-identity-be-understood-as-a-bgg-resolution/27180#27180 Answer by Jim Humphreys for Can the Jacobi-Trudi identity be understood as a BGG resolution? Jim Humphreys 2010-06-05T18:15:47Z 2010-06-05T18:25:01Z <p>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.</p>