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Another interpretation (which is essentially the same as Richard's example of order polynomials if uses order polytopes) is via Ehrhart reciprocity (Stanley Enumerative Combinatorics 1, Section 4.6). …

I think of all of the duality statements you wrote as a consequence of the fact that there is a ring involution of $\Lambda$ sending $e_k$ to $h_k$, so let me give a manifestation of that. First, the …

This is an example in some notes that I worked on. It's a bit involved, and I don't know how to simplify the approach further, so let me just offer a sketch (in particular, I want to try to ignore a b …

There is a new book by Lakshmibai and Raghavan called Standard Monomial Theory which is mostly about how to do invariant theory in a "Schubert varietiesque" way. It is introductory to both Schubert va …

For the exterior powers (highest weight $(1,1,\dots,1)$), the restriction to the orthogonal group is still irreducible -- I think you are forgetting to take $\overline{\lambda} = 0$. There are also …

Background: given a finite-dimensional real vector space V of dimension d, I can define a pointed cone in two ways: either as a set of the form $\{r_1v_1 + \cdots + r_nv_n \mid r_1, \dots, r_n \in R_{ …

The relevance of hom-controlled functors comes from Zwara's paper "Smooth morphisms of module schemes" (Theorem 1.2). The definition there is that two schemes with basepoints $(X,x)$ and $(Y,y)$ have …

Background: For a partition $\lambda$, let $\lambda[N] = (N - |\lambda|, \lambda_1, \lambda_2, \lambda_3, \dots)$, also let $\chi_\lambda$ be the corresponding irreducible character of the symmetric …

The field of definition will be the complex numbers, $V$ is a vector space of dimension $m$, and $O(V)$ is the orthogonal group preserving some nondegenerate bilinear form on $V$. The centralizer alge …

Possibly this is the same thing as Bruce Westbury is saying, but I found this short paper of Kerov, "Realizations of representations of the Brauer semigroup" which constructs the irreducible represent …

Not sure if this is what you're looking for, but the statement for symmetric doubly stochastic matrices is that every such one can be written as a convex combination of $(\sigma + \sigma^t)/2$ where $ …

I'm looking for a "conceptual" explanation to the question in the title. The standard proofs that I've seen go as follows: use the Schubert cell decomposition to get a basis for cohomology and show th …

Embed the hyperoctahedral group $H_n$ into the symmetric group $S_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S_{2n}$ by pa …

Okay I found it in the article by Koike and Terada, "Littlewood's formulas and their application to representations of classical Weyl groups". Here is the theorem. Define $d^\nu_{\lambda, \mu}$ as the …

Let $G$ be reductive group over a field of characteristic $0$ ($GL_n$ fine for this question). Let $V$ be a linear representation of $G$. Then $G$ acts on the tensor algebra $T(V) = \bigoplus_{n \ge 0 …