10
votes
3answers
907 views

A problem on a specific integer partition

Let $n$ be a positive integer, we consider partitions of the following form : $$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that : $d_{i}\vert n$ $1=d_{1}<d_{2} \le d_{3} \le ... \le ...
4
votes
1answer
120 views

Reference for the image of the adjoint to the differential in graph cohomology (which yields STU & IHX)?

One can define cochain complexes of (combinatorial) graphs, where each term is a vector space of linear combinations of certain (isomorphism classes of) graphs, and where the differential $d$ is a ...
16
votes
2answers
419 views

A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion: $$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$ where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
2
votes
1answer
173 views

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 ...
3
votes
0answers
247 views

det(A)det(B) = det(AB+correction), Capelli identities, “factorzied” representation of 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 19-th century, but also ...
11
votes
0answers
530 views

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 ...
4
votes
2answers
294 views

monomials in the universal enveloping of a Lie algebra in terms of the symmetric basis

Let $L$ be a finite-dimensional Lie algebra over a field $k$ of characteristic zero and $e_1,\ldots, e_n$ some basis of $L$. The formula $[e_i,e_j] = \sum_k C_{ij}^k e_k$ determines the structure ...
29
votes
4answers
2k views

Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
3
votes
0answers
194 views

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 ...
7
votes
0answers
185 views

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 ...
10
votes
4answers
635 views

Asymptotics of q-Catalan numbers

q-Catalan numbers are defined recurrently as C0=1, $C_{N+1}=\sum_{k=0}^N q^k C_k C_{N-k}$. What can be said about the asymptotics of Cn when 0<q<1? P.S. In ...
12
votes
2answers
306 views

Derangements and q-variants

Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ derangements of $\{1,2,\dots,n\}$ and that there are $D\_n(q)=(n)_q! \left( ...