All Questions
9 questions
8
votes
0
answers
145
views
Asymptotics of generalized exponents of highest weight modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $H^k$ be the space of homogeneous degree $k$ harmonic polynomials in $\mathrm{Sym}(\mathfrak{g}^*)$ and $H\subset\mathrm{Sym}(\mathfrak{g}^*)...
- 413
6
votes
1
answer
186
views
Action of $\widehat{\mathfrak{sl}_2}$ on symmetric functions with $\mathbb{Z}_{(2)}$ coefficients
It is known that there is a representation of the affine Lie algebra $\widehat{\mathfrak{sl}_q}$ (over $\mathbb{Z}$) on the algebra of symmetric functions, where the action of the Chevalley generators ...
- 3,446
8
votes
0
answers
240
views
Scalar products on symmetric functions behaving like the Macdonald scalar product
The Macdonald symmetric functions (or Macdonald polynomials)
$P_\lambda(x)$ are orthogonal with respect to the Macdonald scalar
product
$$ \langle p_\lambda,p_\mu\rangle =
\delta_{\lambda\mu}z_\...
- 50.8k
5
votes
0
answers
103
views
Form on symmetric functions and their q,t- analogues
[Notations are as in Macdonald's Symmetric Functions and Hall Polynomials]
The space of symmetric functions $\Lambda_{\mathbb{Q}}$ has a bilinear form defined by $ (p_\lambda, p_\mu)= z_\lambda \...
- 820
2
votes
0
answers
99
views
Diagonal operator and infinite wedge space formalism
Let $\bigwedge^{\infty /2}V$ denote semiinfinte wedge space. The followin article section 2 gives a good description about the space and the operator on it.
https://arxiv.org/pdf/math/0207233.pdf
...
- 685
5
votes
0
answers
1k
views
Internal tensor product of strict polynomial functors: is there a more explicit definition?
In the paper Henning Krause, Koszul, Ringel, and Serre duality for strict polynomial functors, arXiv:1203.0311v4, Krause defines something that he calls an "internal tensor product" on the category of ...
- 33.8k
1
vote
1
answer
311
views
Generating function for $t$-residues of partitions using Heisenberg + $\hat{sl_t}$ representation theory
Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let $(n_0,\...
- 1,090
3
votes
1
answer
598
views
An identity for elementary symmetric functions
Trying to understand a result in a representation theoretical paper, I realized that it implies the following elementary identity for symmetric functions. My question is whether this identity is true, ...
- 21.8k
10
votes
1
answer
437
views
Reference request: Grothendieck groups of Hecke algebras at root of unity and symmetric functions
Let $\zeta$ be an $\ell^{\text{th}}$ root of unity, and consider $H_n(\zeta)$, the (finite) Hecke algebra of type A. One can consider a dual pair of Hopf algebras arising from this data, denoted $G(\...
- 485