An abstract for three talks by Nick Kuhn Generic Representation Theory: a survey of basic structure.

With F_{q} a finite field of characteristic p, let F(q) be the category whose objects are functors from finite dimensional F_{q}–vector spaces to F_{q}–vector spaces.

A connection between this abelian category and GL_{n}(F_{q})-modules is easy to see: if F ∈ F(q), then F (F_{q}^{n}) is a GL_{n}(F_{q})-module for all n. The category F(q) blends the categories of GL_{n}(F_{q})-modules together for all n in a very precise way. One is quickly led to lots of naturally occurring infinite length objects of interest.

The three lecturers will survey some of the basic structure of F(q) and related categories. As envisioned, the talks will go roughly as follows.

Lecture 1: Generic representations and the recollment tower.

The category F(q) can be regarded as the category of representations of a small additive category. Thus some basic structure is easy to see: e.g. the projectives and injectives. Two other useful bits of structure are a scalar decomposition and a duality transformation. To see the beginnings of deeper structure, we make two observations.

The first goes as follows. Let GL_{n}(q) and M_{n}(q) denote the categories of modules over the group ring F_{q}[GL_{n}(F_{q})] and the semigroup ring F_{q}[M_{n}(F_{q})]. Then F(q) is equivalent to the inverse limit of the tower of categories

… ⟶ M_{n+1}(q) ⟶ M_{n}(q) ⟶ M_{n−1}(q) ⟶ …

and each ‘fiber sequence’

GL_{n}(q) ⟶ M_{n}(q) ⟶ M_{n−1}(q)

is part of a ‘recollement diagram’. This observation has many consequences. For example, it follows formally that simple objects in F(q) correspond to the simple GL_{n}(F_{q})-modules, for all n.

The second observation is that the simple-minded operation of direct sum in the category of finite dimensional F_{q}-vector spaces induces all sorts of interesting structure on F(q). For example, this leads to the notion of a polynomial functor in the sense of Eilenberg-MacLane, which leads to more recollement diagrams involving the symmetric groups. Another consequence is a remarkable Kunneth formula for computing many Ext groups.

Lecture 2: Finite functors and the role of the symmetric powers.

Using the notion of a polynomial functor, it is easy to characterize the finite objects, and then the locally finite objects. An unresolved (and perhaps misnamed) Artinian Conjecture goes as follows: F(q) is a locally Noetherian category. (See Geoff Powell’s talk for more about this.)

We then turn our attention to two (dual) families of finite objects in F(q): S_{n} and S^{n}, n ≥ 0, defined by S_{n}(V) = (V^{⊗n})^{Σ_{n}}, and S^{n}(V) = (V^{⊗n})_{Σ_{n}}. Two fundamental theorems are the Embedding Theorem which says that every finite object embeds in a sum of S^{n}’s, and the Vanishing Theorem which says that S^{n}, when localized by inverting the Frobenious map, becomes injective in the category of locally finite objects.

Via the Gabriel–Popescu Theorem, these theorems translate into statements about the representation category U(q) = Rep(S^{∗}). When q is a prime, the category U(q) is revealed to be the category of unstable modules over the Steenrod algebra: a category that has been of interest to algebraic topologists for roughly half a century!

Lecture 3: Ext calculations, K-theory, and rational cohomology.

In this talk, I will survey work of Franjou-Lannes-Schwartz, Franjou-Friedlander-Skorichenko-Suslin, Betley, Pirashvili, myself, and others concerning calculations of Ext groups in F(q), and how these Ext groups are related to algebraic K–theory, Topological Hochshild Cohomology, finite group cohomology, and rational cohomology.

References

1. N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: I, Amer. J. Math. 116(1994), 327–360.

2. N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: II, K-Theory J. 8(1994), 395–428.

3. N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: III, K-Theory J. 9(1995), 273–303.

4. N. J. Kuhn, Rational cohomology and cohomological stability in generic representation theory, preprint 1997, Amer.J.Math, to appear.

5. H.-W. Henn, J. Lannes, and L. Schwartz, The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects, Amer. J. Math. 115(1993), 1053–1106.

6. V. Franjou, J. Lannes, and L. Schwartz, Autor de la cohomologie de MacLane des corps finis, Inv. Math. 115(1994), 513–538.

7. E. M. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Inv. Math. 127(1997), 209–270.

8. V. Franjou, E. M. Friedlander, A. Scorichenko, and A. Suslin, General linear and functor cohomology over finite fields, preprint, 1997.