All Questions
7 questions
8
votes
0
answers
488
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det(A)det(B) = det(AB+correction), Capelli identities, "factorized" representation of $\mathfrak {gl}_n$
Context: Some probably know that there are Capelli identities which state
$$det(A)det(B) = det(AB+correction)$$ for some matrices with non-commuting elements, they go back to the 19-th century, but ...
- 13.4k
2
votes
0
answers
87
views
Modules over quantum complete intersections
Let $a_i \geq 2$ be natural numbers and $q_{ij}$ field elements of the field $k$ for $i>j$.
A quantum complete intersection is the algebra $A:=k<x_1,...,x_n>/(x_i^{a_i},x_i x_j - q_{ij} x_j ...
- 26.5k
20
votes
1
answer
586
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$q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?
The following formula of astonishing beauty and power (imho):
$$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...
- 14.6k
3
votes
0
answers
220
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Generalisation of the quantum exterior algebra
One might generalise the classical exterior algebra as follows to the quantum exterior algebra:
$K<x_1,...x_n>/(x_i^2,x_i x_j + q_{i,j}x_j x_i)$ with nonzero field elements $q_{i,j}$ for $i<j$...
- 26.5k
2
votes
1
answer
228
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A question on Lusztig's `graph with automorphism' construction?
Using the notion of a graph with compatible automorphism, Lusztig constructs all symmetrizable Cartan data (i.e. Cartan matrices $A$ for which there is a diagonal matrix $D=\mathrm{diag}(d_1,\ldots,...
- 44.7k
16
votes
0
answers
824
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Capelli determinant = Duflo ( determinant) - was it known ?
Question briefly. Was this fact known: Capelli determinant = Duflo (determinant) ? (This is an equality of the two central elements in universal enveloping of Lie algebra $gl_n$).
I googled a lot ...
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- 1
7
votes
0
answers
213
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Decomposition of certain projectives for cyclotomic q-Schur algebras
In representation theory, a very popular set of finite dimensional algebras are the $q$-Schur algebras, which are given by looking at the endomorphisms of $V^{\otimes d}$ where $V$ is the standard ...
- 44.7k