A numerical invariant for linear representations of finite groups

Abstract

Abstract. We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized Severi-Brauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type, extending earlier results of the first author. As a consequence, we prove analogues of classical theorems of R. Brauer and O. Schilling about the Schur index, where the Schur index of a representation is replaced by its essential dimension. In the last section we show that essential dimension of representations can behave in rather unexpected ways in the modular setting.

Fields of Definition for Representations of Associative Algebras

“…Then Proof. (a) The assertion of part (a), follows from [KRP,Example 6.1]. For the sake of completeness, we will give an independent proof.…”

Incompressible Grassmannians of isotropic subspaces

“…If φ is isotropic, the variety X i is equivalent (in the above sense of existence of rational maps in both directions) to the variety X ′ i−1 . Therefore cdim 2 X i = cdim 2 X ′ i−1 (see, e.g., [12,Lemma 3.3]). Since dim X i > dim X ′ i−1 , we get a contradiction with the 2-incompressibility of X i .…”

Unitary Grassmannians of Division Algebras

“…In particular, canonical 2-dimensions of these two varieties coincide (see e.g. [15,Lemma 3.3]). Hence {l 1 , .…”

“…For any i, the variety X i is a closed subvariety of the Weil transfer R K/F SB i D, where SB i D is the ith generalized Severi-Brauer variety of D -the K-variety of all right ideals in D of reduced dimension i. We recall that according to [10] (see [15] for a more recent and simple proof), for any r = 0, 1, . .…”

“…We recall that according to [10] (see [15] for a more recent and simple proof), for any r = 0, 1, . .…”