Stiefel-Whitney invariants for bilinear forms

Uriya A. First; Uzi Vishne

doi:10.1016/j.laa.2013.05.024

Stiefel-Whitney invariants for bilinear forms

Uriya A. First, Uzi Vishne

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

Abstract

We examine potential extensions of the Stiefel-Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt's Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)

Original language	English
Pages (from-to)	1905-1917
Number of pages	13
Journal	Linear Algebra and Its Applications
Volume	439
Issue number	7
DOIs	https://doi.org/10.1016/j.laa.2013.05.024
State	Published - 1 Oct 2013

Bibliographical note

Funding Information:
✩ This work was supported by the U.S.–Israel Binational Science Foundation (grant no. 2010/149).

Funding

✩ This work was supported by the U.S.–Israel Binational Science Foundation (grant no. 2010/149).

Funders	Funder number
U.S.-Israel Binational Science Foundation	2010/149

Keywords

Bilinear form
Chain lemma
Milnor's K-ring
Quadratic form
Stiefel-Whitney invariant
Upper-triangular matrix

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10.1016/j.laa.2013.05.024

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AB - We examine potential extensions of the Stiefel-Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt's Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)

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KW - Chain lemma

KW - Milnor's K-ring

KW - Quadratic form

KW - Stiefel-Whitney invariant

KW - Upper-triangular matrix

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Stiefel-Whitney invariants for bilinear forms

Abstract

Bibliographical note

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Keywords

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