Timeline for What are the matrices preserving the $\ell^1$-norm?
Current License: CC BY-SA 3.0
11 events
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| Apr 7, 2022 at 21:03 | comment | added | Hans | I am reading up on "absolutely irreducibility" and the bilinear invariance. At the same time, would you care to answer this question math.stackexchange.com/q/4418122/64809, where I query about a different proof in a paper for this very same theorem which I cannot make sense of? | |
| Apr 4, 2022 at 22:10 | comment | added | YCor | @Hans I should have said "absolutely irreducibly". This is a general fact that for an absolutely irreducible representation with an invariant nondegenerate bilinear form, the space of invariant bilinear forms has dimension 1. | |
| Apr 4, 2022 at 18:01 | comment | added | Hans | Thank you, YCor. Would you please explain why 1) "since $W$ acts irreducibly on $\mathbf{R}^n$, all scalar products it preserves are collinear."? 2) "Since $G$ is compact, it preserve a scalar product, and hence since $W\subset G$ we deduce $G\subset\mathrm{O}(n)$."? | |
| Apr 2, 2022 at 10:31 | comment | added | YCor | @Hans (a) I mean that for any two scalar products preserved by $G$, one is a (positive) scalar multiple of the other. (b) yes O($n$) is the orthogonal group (c) if $K_p$ is the $\ell^p$ unit sphere I mean that $K_2\cap K_p$ is $G$-invariant (since both $K_2$ and $K_p$ are $G$-invariant). | |
| Apr 2, 2022 at 0:51 | comment | added | Hans | This looks great. But the information is a bit dense. Would you please be so kind as to unpack and expand out the details or provide references for the intermediate propositions? 1) Specifically, could you please write out the derivation for paragraph 2, e.g., what you mean by "scalar products are colinear"? Is $O(n)$ the orthogonal group? 2) Why does "Hence $G$ preserve the intersection of the $\ell^2$ and $\ell^p$ unit spheres"? | |
| Dec 10, 2017 at 14:16 | comment | added | YCor | My initial approach consisted in proving that $W$ is a maximal subgroup of $\mathrm{O}(n)$ for $n\ge 3$ (it's false for $n=2$)- I mention it as it yields a stronger conclusion. It's easy to see that $W$ acts irreducibly for the adjoint representation, so any proper subgroup containing $W$ is finite. Using the classification of groups generated by reflections (see standard treatments on Coxeter groups), $W(\simeq B_n)$ is maximal among finite subgroups with the possible exceptions $n=4,8$, which I haven't checked properly (need to discard inclusions $B_4\subset F_4$ and $B_8\subset E_8$). | |
| Dec 9, 2017 at 19:03 | comment | added | YCor | Thanks (I erased a draft longer message since I initially started using some Lie group stuff, and then forgot copying the definition of $W$, which could be guessed by people familiar with Weyl groups, or just because it's accidentally defined in the message as the stabilizer of $V$) | |
| Dec 9, 2017 at 19:01 | history | edited | YCor | CC BY-SA 3.0 |
added 44 characters in body
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| Dec 9, 2017 at 19:00 | comment | added | Denis Chaperon de Lauzières | Signed permutations of the canonical basis vectors -- it is a priori contained in the isometry group. | |
| Dec 9, 2017 at 18:56 | comment | added | j.c. | I may have missed this somewhere, but what is W? | |
| Dec 9, 2017 at 18:47 | history | answered | YCor | CC BY-SA 3.0 |