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Results tagged with finite-groups
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user 1306
Questions on group theory which concern finite groups.
7
votes
3
answers
435
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Subgroups of $GL(k,q)$ for bounded $k$
This question on subgroups of $GL(2,q)$ asked by Jan, and especially wonderful answers to it given by Geoff Robinson, Ralph, and Will Sawin showing that "almost no finite groups" inject in $GL(2,q)$ m …
- 16.9k
1
vote
Generate harmonic polynomials for a finite group
Maybe you need to be more clear about the word "similar". For example, the celebrated case of "diagonal harmonics" corresponding to the diagonal action of $S_n$ on $(\mathbb{C}^{2})^n$ there is some r …
- 16.9k
15
votes
Accepted
Reference for this theorem in representation theory?
I am not quite sure about the reference :( I always thought of this fact as follows.
Matrix elements of tensor powers of a representation U are all possible monomials in matrix elements of U, so the …
- 16.9k
7
votes
1
answer
560
views
recognition of symmetric groups in GAP
In GAP (https://www.gap-system.org), there is a function IsSymmetricGroup, which tells you whether a subgroup of $S_n$ generated by given permutations is all of the $S_n$. It looks like it takes virtu …
- 16.9k
7
votes
factorization of the regular representation of the symmetric group
The fact that the Lie module (as proposed by Darij Grinberg) works, as well, as an explicit isomorphism of modules, follows from the theory of cyclic operads: see Corollary 6.9 in http://sites.math.no …
- 16.9k
3
votes
Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$
If I remember correctly, one of the symmetric powers should become divided powers. A quick google search brought me to Section 3.4 of Aprodu, Farkas, Papadima, Raicu, and Weyman - Koszul modules and G …
- 16.9k
46
votes
Fun applications of representations of finite groups
An example from Kirillov's book on representation theory: write numbers 1,2,3,4,5,6 on the faces of a cube, and keep replacing (simultaneously) each number by the average of its neighbours. Describe ( …