Timeline for Gaussian elimination is just Gram-Schmidt with a change to the inner product symbol?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 1, 2023 at 15:05 | comment | added | wlad | My chain complex of quadratic forms is a flag equipped with some additional structure. I arrived at it by noticing that in the $2\times2$ case, you can use quadratic forms without flags. I don't know if it's a helpful idea or not, though. | |
| Mar 1, 2023 at 14:29 | history | edited | Callum | CC BY-SA 4.0 |
missing < and a few commas
|
| Mar 1, 2023 at 14:18 | history | edited | LSpice | CC BY-SA 4.0 |
Minor proofreading
|
| Feb 26, 2023 at 20:02 | comment | added | LSpice | I think the appropriate framework for a general quasi-split, reductive group $G$ to replace most talk of bases is pairs of a Borel $B$ and a maximal torus $T$ in $B$; for $\operatorname{GL}_{n,k}$, these parameterise "ordered, projective bases", i.e., decompositions of $k^n$ as a direct sum of lines, where the order of the summands matters. (As you indicate, we do need to consider bases as ordered, and not just as sets, in order to make sense of, say, upper-triangularity.) As with bases for $\operatorname{GL}_n$, two such objects are $G(k)$-conjugate, but slightly less uniquely (up to $T(k)$). | |
| Feb 26, 2023 at 17:18 | comment | added | LSpice | (3rd try's a charm, I hope!) More generally, in the LUP decomposition, the P records to which Bruhat cell you belong, in the following sense. If $w_0^{-1} g w_0$ belongs to $B w B$, where $w_0 = \operatorname{antidiag}(1, \dotsc, 1)$, then we can write $w_0^{-1} g w_0 = b w u$ for $w$ a permutation matrix and $u$ in a suitable subgroup $U_w$ depending on $w$; and then $g$ equals $(w_0 b w_0^{-1})(w_0 w u w^{-1}w_0^{-1})w_0 w$, with $w_0 b w_0^{-1}$ lower-triangular, $w_0 w u w^{-1}w_0^{-1}$ upper triangular, and $w_0 w$ a permutation matrix. | |
| Feb 26, 2023 at 15:34 | history | answered | Callum | CC BY-SA 4.0 |