Incompressibility of quadratic Weil transfer of generalized Severi–Brauer varieties

Karpenko, Nikita A.

doi:10.1017/s1474748011000090

Cited by 9 publications

(7 citation statements)

References 16 publications

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“…We use this to show that ed(ρ) ≤ |G| 2 /4 for any n, k, and K/k in Section 9 and to prove a variant of a classical theorem of Brauer in Section 10. In Section 11 we compute the canonical dimension of a broad class of Weil transfers of generalized Severi-Brauer varieties, extending earlier results of the first author from [18] and [20]. This leads to a formula for the essential p-dimension of an irreducible character in terms of its decomposition into absolutely irreducible components; see Corollary 12.3.…”

Section: Introductionmentioning

confidence: 77%

A numerical invariant for linear representations of finite groups

Karpenko¹,

Reichstein²

2015

Comment. Math. Helv.

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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.

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Section: Introductionmentioning

confidence: 77%

A numerical invariant for linear representations of finite groups

Karpenko¹,

Reichstein²

2015

Comment. Math. Helv.

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“…The proof proceeds by induction on k. Let us do the induction base k = 0. According to [12,Theorem 1.2], U(X 0 ) = M(X 0 ). Since rk M (X(1, D)) = 2 n , it follows from Lemma 5.1 that rk F U(X 0 ) = 2 n , rk K U(X 0 ) = 2 n−1 (2 n − 1).…”

Section: Some Ranks Of Some Motivesmentioning

confidence: 99%

“…By [10] and [12], M is a shift of the motive U(X l ) with some l ∈ [0, k − 1] or a shift of the motive cor K/F U(X (2 l , D)) with some l ∈ [0, k]. In the first case we are done by the induction hypothesis.…”

Section: Some Ranks Of Some Motivesmentioning

confidence: 99%

Hyperbolicity of unitary involutions

Karpenko

2012

Sci. China Math.

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Abstract. We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.

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

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confidence: 99%

“…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, . .…”

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confidence: 99%

Unitary Grassmannians of Division Algebras

Karpenko

2015

Transformation Groups

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Abstract. We consider a central division algebra over a separable quadratic extension of a base field endowed with a unitary involution and prove 2-incompressibility of certain varieties of isotropic right ideals of the algebra. The remaining related projective homogeneous varieties are shown to be 2-compressible in general. Together with [13], where a similar issue for orthogonal and symplectic involutions has been treated, the present paper completes the study of grassmannians of isotropic ideals of division algebras.Let F be a field, K/F a separable quadratic field extension, n an integer ≥ 1, and D a central division K-algebra of degree 2 n endowed with a K/F -unitary involution σ. For definitions as well as for basic facts about involutions on central simple algebras, we refer to [16].For any integer i, we write X i for the F -variety of isotropic (with respect to) For any i, the variety X i is smooth and projective. It is nonempty if and only if 0 ≤ i ≤ 2 n−1 (X 0 is simply Spec F ) in which case it is geometrically connected and has dimensionFor 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, . . . , n − 1, the variety R K/F SB 2 r D is 2-incompressible. This means, roughly speaking, that any self-correspondenceof odd multiplicity is dominant. In particular, any rational self-mapThe following theorem is the main result of this note. It extends to the unitary setting the results on orthogonal and symplectic involutions obtained in [13]. Theorem 1. For any r = 0, 1, . . . , n − 1, the variety X 2 r is 2-incompressible.

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Incompressibility of quadratic Weil transfer of generalized Severi–Brauer varieties

Cited by 9 publications

References 16 publications

A numerical invariant for linear representations of finite groups

A numerical invariant for linear representations of finite groups

Hyperbolicity of unitary involutions

Unitary Grassmannians of Division Algebras

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