Abstract
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.
Highlights
It could be that the irreducibles of each Gd are parameterized in some common manner, and representations of different G are considered compatible if they have matching irreducible constituents. (This definition is too weak in practice, but a good first approximation.) The stable representation theory c The Author(s) 2015
There are two natural questions to consider. (A) What is the structure of the stable representation theory of G? To be more precise, we introduce the stable representation category Repst(G∗): informally, its objects are compatible sequences of representations, with two such sequences considered equivalent if they are the same for d large; formally, it can be described as a Serre quotient
We study the stable representation theory of five families of groups: the three families of classical groups GL(d), O(d) and Sp(d), the symmetric groups S(d), and the general affine groups GA(d)
Summary
We obtain an equivalence between the category Rep(S∗) of sequences (Mn)n 0, where Mn is a representation of Sn, and the category Reppol(GL) This provides a combinatorial description of the stable polynomial representation theory of GL. We obtain a covariant equivalence by using the upwards Brauer category instead This is our combinatorial description of the stable representation theory of the orthogonal group. The previous paragraph can be rephrased in this language as follows: the category Rep(O) is equivalent to the category of finite length modules over the tca Sym(Sym2(V)) This is our commutative algebraic description of the stable representation theory of the orthogonal group. There are likely many other changes to be made, but we have not thought through the details
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