| Mon, Nov 16, 2026
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Graduate Student Seminar
4:00 PM
MSCS 514 | | A Decomposition for Pairs of Bilinear Forms
Emily Franklin, OSU
| | | Abstract: Over a field of characteristic not 2, the general linear group $\operatorname{GL}(n)$ acts on the direct sum $\operatorname{AMat}(n) \oplus \operatorname{SMat}(n)$ by $g \cdot (A, B) = (g A g^{T}, g B g^{T}).$ Though Kronecker and Dickson gave a classification of orbits of this action via matrix methods, Waterhouse initiated a module-theoretic treatment in 1976 for pairs of symmetric forms, $\operatorname{SMat}(n) \oplus \operatorname{SMat}(n),$ by studying finite-dimensional vector spaces equipped with two quadratic forms. The same year, Scharlau gave a parallel treatment for pairs of alternating forms. In this talk, we adopt this algebraic perspective to address the ``mixed'' case. We establish a decomposition for finite-dimensional vector spaces equipped with both an alternating bilinear form and a symmetric bilinear form, yielding a cohesive, module-theoretic framework that simultaneously encompasses $\operatorname{SMat}(n) \oplus \operatorname{SMat}(n),$ $\operatorname{AMat}(n) \oplus \operatorname{AMat}(n),$ and $\operatorname{SMat}(n) \oplus \operatorname{AMat}(n).$ |
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Mathematics Seminars and Colloquia