Timeline for Conceptual explanation for curious linear-algebra fact in characteristic $2$

9 events

when toggle format	what		by	license	comment
Dec 13, 2018 at 20:38	vote	accept	Alice
Dec 13, 2018 at 1:35	comment	added	David E Speyer		@user44191 It's a nontrivial extension, not a direct sum. The diagonal zero symmetric matrices are a subspace and the "diagonal only" are a quotient.
Dec 12, 2018 at 23:59	comment	added	spin		Perhaps it is better to express this in terms of vector spaces: $X$ corresponds to a symmetric bilinear form $b$ on a vector space $V$ over $\mathbb{F}_2$, and $X_0$ corresponds to the linear map $v \mapsto b(v,v)$. Further $A^tXA$ corresponds to the bilinear form $b'$ defined by $b'(v,w) = b(Av, Aw)$, with the associated linear map $v \mapsto b'(v,v)$ etc. Looking at it this way, the equality seems clear.
Dec 12, 2018 at 23:44	answer	added	dhy		timeline score: 6
Dec 12, 2018 at 23:27	history	edited	user44191		Retagged to the specific characteristic, added rt for reasons explained in comment
Dec 12, 2018 at 23:27	comment	added	user44191		This likely can be seen as a statement about the representation theory of $G = SL_n(\mathbb{F}_2)$ over $\mathbb{F}_2$. If $\rho$ is the obvious representation, then this says that the symmetric component of $\rho \otimes \rho^t$ further decomposes to a diagonal-only component and a zero-diagonal component.
Dec 12, 2018 at 22:10	history	edited	Alice	CC BY-SA 4.0	edited body
Dec 12, 2018 at 21:40	review	First posts
Dec 12, 2018 at 21:48
Dec 12, 2018 at 21:37	history	asked	Alice	CC BY-SA 4.0