Explore the vast range of titles available.

This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure.

We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results: for example, the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, and a canonical derived auto-equivalence of the general linear theory.

Twisted commutative algebras (tca’s) have played an important role in the nascent field of representation stability. Let$A_{d}$be the tca freely generated by$d$indeterminates of degree 1. In a previous paper, we determined the structure of the category of$A_{1}$-modules (which is equivalent to the category of$\\mathbf{FI}$-modules). In this paper, we establish analogous results for the category of$A_{d}$-modules, for any$d$. Modules over$A_{d}$are closely related to the structures used by the authors in previous works studying syzygies of Segre and Veronese embeddings, and we hope the results of this paper will eventually lead to improvements on those works. Our results also have implications in asymptotic commutative algebra.

We show that for Jack parameter α = −( k + 1)/( r − 1), certain Jack polynomials studied by Feigin–Jimbo–Miwa–Mukhin vanish to order r when k + 1 of the coordinates coincide. This result was conjectured by Bernevig and Haldane, who proposed that these Jack polynomials are model wavefunctions for fractional quantum Hall states. Special cases of these Jack polynomials include the wavefunctions of Laughlin and Read–Rezayi. In fact, along these lines we prove several vanishing theorems known as clustering properties for Jack polynomials in the mathematical physics literature, special cases of which had previously been conjectured by Bernevig and Haldane. Motivated by the method of proof, which in the case r = 2 identifies the span of the relevant Jack polynomials with the S n -invariant part of a unitary representation of the rational Cherednik algebra, we conjecture that unitary representations of the type A Cherednik algebra have graded minimal free resolutions of Bernstein–Gelfand–Gelfand type; we prove this for the ideal of the ( k + 1)-equals arrangement in the case when the number of coordinates n is at most 2 k + 1. In general, our conjecture predicts the graded S n -equivariant Betti numbers of the ideal of the ( k + 1)-equals arrangement with no restriction on the number of ambient dimensions.

Given a category C\\mathcal {C} of a combinatorial nature, we study the following fundamental question: how do combinatorial properties of C\\mathcal {C} affect algebraic properties of representations of C\\mathcal {C}? We prove two general results. The first gives a criterion for representations of C\\mathcal {C} to admit a theory of Gröbner bases, from which we obtain a criterion for noetherianity. The second gives a criterion for a general “rationality” result for Hilbert series of representations of C\\mathcal {C}, and connects to the theory of formal languages. 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 (studied by Church, Ellenberg, and Farb), and certain generalizations, are noetherian; we prove the Lannes–Schwartz artinian conjecture from the study of generic representation theory of finite fields; we significantly improve the theory of Δ\\Delta-modules, introduced by Snowden in connection to syzygies of Segre embeddings; and we establish fundamental properties of twisted commutative algebras in positive characteristic.

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group GG. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible GG-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly. Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellenberg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable.

For a field$\\text{k}$, we prove that the$i$th homology of the groups$\\operatorname{GL}_{n}(\\text{k})$,$\\operatorname{SL}_{n}(\\text{k})$,$\\operatorname{Sp}_{2n}(\\text{k})$,$\\operatorname{SO}_{n,n}(\\text{k})$, and$\\operatorname{SO}_{n,n+1}(\\text{k})$with coefficients in their Steinberg representations vanish for$n\\geqslant 2i+2$.

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of Špenko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves of its various geometric invariant theory (GIT) quotients for suitably generic stability parameters. These variations of GIT quotient are examples of more complicated wall crossings than the balanced wall crossings studied in recent work on derived categories and variation of GIT quotients. Our construction is algorithmic and quite explicit, allowing us to: 1) describe a tilting vector bundle which generates the derived category of such a GIT quotient, 2) provide a combinatorial basis for the K-theory of the GIT quotient in terms of the representation theory of G, and 3) show that our derived equivalences satisfy certain relations, leading to a representation of the fundamental groupoid of a “Kähler moduli space” on the derived category of such a GIT quotient. Finally, we use graded categories of singularities to construct derived equivalences between all Deligne–Mumford hyperkähler quotients of a symplectic linear representation of a reductive group (at the zero fiber of the algebraic moment map and subject to a certain genericity hypothesis on the representation), and we likewise construct actions of the fundamental groupoid of the corresponding Kähler moduli space.

Over a field of characteristic zero, we prove that for each r , there exists a constant C ( r ) so that the prime ideal of the r th secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most C ( r ). The main idea is to consider the coordinate ring of all of the ambient spaces of the Veronese embeddings at once by endowing it with the structure of a Hopf ring, and to show that its ideals are finitely generated. We also prove a similar statement for partial flag varieties and, in fact, arbitrary projective schemes, and we also get multi-graded versions of these results.