All Questions
Tagged with qa.quantum-algebra co.combinatorics
11 questions with no upvoted or accepted answers
16
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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 ...
- 24.9k
8
votes
0
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488
views
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 ...
- 24.9k
7
votes
0
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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
5
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154
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Interpretation of superfactorial in terms of plane partitions
Recently I got interested in plane partitions and the following formula by MacMahon, which counts the number of plane partitions $\pi \in B(r,s,t)$ fitting in an $(r,s,t)$-box:
$$
\binom{r+s+t}{r,s,t}...
- 1,813
5
votes
0
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345
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A fusion ring identity
Fusion rings
I'll more or less stick to the presentation given in this question: [1]
We define a fusion ring as follows: consider a free $\mathbb{Z}$-module $\mathbb{Z}\mathcal{B}$ with finite basis ...
- 309
3
votes
0
answers
97
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What algebras generate polynomial count varieties as their representations spaces ? Is it preserved by the Koszul duality, Manin's endomorphisms?
Consider for example commutative polynomial algebra $C[x,y]: xy=yx$, look on $F_p$-matrices satisfying that relation - the number over $F_p$ will be given by polynomial in $p$ (classical result due to ...
- 24.9k
3
votes
0
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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
3
votes
0
answers
216
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A "nice" Orthogonal Basis for Translation Invariant Symmetric Polynomials
It is going to be a rather long question, so I will first state it and then try to explain and motivate it.
Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
- 613
3
votes
0
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233
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How many set partitions on a big cube’s boundary arise from cubomino decompositions of the solid cube?
Introduction. This is a counting question about configurations that can appear on the outside of assembled Soma cube-like puzzles. More specifically, it’s about the ways in which the pieces of an ...
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2
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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
0
votes
0
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167
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Negative $q$-binomial series: reference request
There seems to be a result for formal series (I hope this is right) for all integer $r\ge 0$
$$
\sum_{n\ge 0} (-x)^n\ {{n+r}\choose{r}}_{q} = (1+x)^{-1}(1+qx)^{-1}\dots
(1+q^{r}x)^{-1}
$$
where the $q$...
- 1,143