2,679 research outputs found

### Orthosymplectic Lie superalgebras, Koszul duality, and a complete intersection analogue of the Eagon-Northcott complex

We study the ideal of maximal minors in Littlewood varieties, a class of
quadratic complete intersections in spaces of matrices. We give a geometric
construction for a large class of modules, including all powers of this ideal,
and show that they have a linear free resolution over the complete intersection
and that their Koszul dual is an infinite-dimensional irreducible
representation of the orthosymplectic Lie superalgebra. We calculate the
algebra of cohomology operators acting on this free resolution. We prove
analogous results for powers of the ideals of maximal minors in the variety of
length 2 complexes when it is a complete intersection, and show that their
Koszul dual is an infinite-dimensional irreducible representation of the
general linear Lie superalgebra.
This generalizes work of Akin, J\'ozefiak, Pragacz, Weyman, and the author on
resolutions of determinantal ideals in polynomial rings to the setting of
complete intersections and provides a new connection between representations of
classical Lie superalgebras and commutative algebra. As a curious application,
we prove that the cohomology of a class of reducible homogeneous bundles on
symplectic and orthogonal Grassmannians and 2-step flag varieties can be
calculated by an analogue of the Borel-Weil-Bott theorem.Comment: 35 pages; v2: updated reference

### Gr\"obner methods for representations of combinatorial categories

Given a category C of a combinatorial nature, we study the following
fundamental question: how does the combinatorial behavior of C affect the
algebraic behavior of representations of C? We prove two general results. The
first gives a combinatorial criterion for representations of C to admit a
theory of Gr\"obner bases. From this, we obtain a criterion for noetherianity
of representations. The second gives a combinatorial criterion for a general
"rationality" result for Hilbert series of representations of C. This criterion
connects to the theory of formal languages, and makes essential use of results
on the generating functions of languages, such as the transfer-matrix method
and the Chomsky-Sch\"utzenberger theorem.
Our work is motivated by recent work in the literature on representations of
various specific categories. Our general criteria recover many of the results
on these categories that had been proved by ad hoc means, and often yield
cleaner proofs and stronger statements. For example: we give a new, more
robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb),
and a family of natural generalizations, are noetherian; we give an easy proof
of a generalization of the Lannes-Schwartz artinian conjecture from the study
of generic representation theory of finite fields; we significantly improve the
theory of $\Delta$-modules, introduced by Snowden in connection to syzygies of
Segre embeddings; and we establish fundamental properties of twisted
commutative algebras in positive characteristic.Comment: 41 pages; v2: Moved old Sections 3.4, 10, 11, 13.2 and connected text
to arxiv:1410.6054v1, Section 13.1 removed and will appear elsewhere; v3:
substantial revision and reorganization of section

### Hilbert series for twisted commutative algebras

Suppose that for each n >= 0 we have a representation $M_n$ of the symmetric
group S_n. Such sequences arise in a wide variety of contexts, and often
exhibit uniformity in some way. We prove a number of general results along
these lines in this paper: our prototypical theorem states that if $M_n$ can be
given a suitable module structure over a twisted commutative algebra then the
sequence $M_n$ follows a predictable pattern. We phrase these results precisely
in the language of Hilbert series (or Poincar\'e series, or formal characters)
of modules over tca's.Comment: 28 page

### VIC-modules over noncommutative rings

For a finite ring $R$, not necessarily commutative, we prove that the
category of $\text{VIC}(R)$-modules over a left Noetherian ring $\mathbf{k}$ is
locally Noetherian, generalizing a theorem of the authors that dealt with
commutative $R$. As an application, we prove a very general twisted homology
stability for $\text{GL}_n(R)$ with $R$ a finite noncommutative ring.Comment: 20 page

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