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Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

4 votes
0 answers
98 views

Is there a cohomological interpretation of the bilinear form arising from Clifford theory?

For this question all groups are finite, and representations are over $\mathbb{C}$. The setup is that we have $N$ a normal subgroup of $G$, with abelian quotient $A$, and an irrep $V$ of $N$ that is i …
Chris H's user avatar
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21 votes
0 answers
468 views

Is there a "direct" proof of the Galois symmetry on centre of group algebra?

Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$ This induces a linear automorphi …
Chris H's user avatar
  • 1,949
11 votes
1 answer
215 views

A question about the adjoint of the Adams operations on representation rings

Let $G$ be a finite group, and $R(G)$ its representation ring over $\mathbb{C}$. We have the Adams operations $\psi^k:R(G)\rightarrow R(G)$, given on the level of characters by: $$\chi_{\psi^k{V}}(g)= …
Chris H's user avatar
  • 1,949
3 votes
0 answers
78 views

"Character" theory via dualisable $2$ categories

One interesting way to describe the ordinary (over $\mathbb{C}$) character theory of finite groups is to view the categories $Rep(G)$ together in a $2$ category with bimodules as morphisms. This $2$ c …
Chris H's user avatar
  • 1,949
1 vote
0 answers
97 views

Do you recognise this setup of structure on a poset?

The setup is that we have a finite poset $P$, with a multiplicative rank function $r_{xy}:P\times P\rightarrow \mathbb{N}$, and a symmetric pairing $\langle\ ,\ \rangle:P\times P\rightarrow\mathbb{N}$ …
Chris H's user avatar
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6 votes
1 answer
291 views

What properties of a finite group fibre functor give its endomorphisms a hopf algebra struct...

Tannaka duality for a finite group lets us recover the group algebra $\mathbb{C}[G]$ as the endomorphisms of the forgetful functor $F:RepG\rightarrow Vect$, and taking the monoidal automorphisms recov …
Chris H's user avatar
  • 1,949
5 votes
2 answers
424 views

Is the Perron-Frobenius dimension of a G-Set given by its cardinality?

Given a ring $R$ with finite additive basis $\{e_i\}_{i=1}^{n}$, such that $e_i e_j=\sum c_{ijk}e_k$ with $c_{ijk}\in \mathbb{N}$, we define the Perron-Frobenius dimension $FPDim(e_i)$ of a basis elem …
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