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3 additional info

I've gotten stuck in a project I have been working on, essentially on the following combinatorial question about the symmetric group.

I am interested in this because it may allow a way to directly interpolate the construction of these modules to complex rank, via Pavel Etingof's program. (I believe one can interpolate the construction in another way, by reasoning more directly on the definition of the category $Rep(S_t)$ given by Deligne, but this seems to be useless as far as explicit computations--which might be helpful to study the "degeneracy phenomena" that Etingof has suggested might exist--are concerned.) In this case we have tensor categories $Rep(S_t)$ for $t$ not necessarily an integer, and while the interpretation in terms of vector spaces fails, the one in terms of Young diagrams does not.

Edit (12/27) I added a bounty today and here is some additional information that may be useful:

It should come out that the matrices representing the $y$-morphism $\mathfrak{h} \otimes M^{\pi}_c \to M^{\pi}_c$ are polynomials in the dimension $n$. When increasing $n$, we change the partition $\pi$ by adding to the first (largest) element and leaving the rows below fixed. Since a simple object in $Rep(S_t)$ for $t$ not an integer can be represented as a normal Young diagram (of size, say, $N$) with a "very long line" of "size" $t-N$ at the top, this kind of a polynomial interpolation will allow for an interpolation of the $M^{\pi}_c$ to complex rank. My claim that it should come out as a polynomial was based upon studying induction directly on these categories and finding it was interpolable. However, I don't know how to compute the $y$'s directly as matrices via the simple object decomposition.My hope was that there is a clean not-too-computationally-intensive way to do this, but unfortunately I'm not yet sufficiently comfortable with the theory of the symmetric group to have any ideas as to how to proceed.

I am also interested in the degenerate affine Hecke algebra of type A, where these kinds of standard induced modules can be defined similarly. Their simple quotients form a complete collection of irreducible modules for the Hecke algebra according to a theorem of Zelevinsky, and I know that these, too, can be interpolated by reasoning on the definitions in Deligne's paper (so one gets objects in the interpolated category $Rep(H_t)$, which is defined in Etingof's talk). But I am curious here too how it is possible to compute the $y$-morphisms as matrices using the decomposition into irreducibles in $Rep(S_t)$.

2 Fix a display

I've gotten stuck in a project I have been working on, essentially on the following question.

One can obtain a 1-dimensional representation $M^n_c$ of the algebra $T_n := S_n \rtimes \mathbb{C}[y_1, \dots, y_n]$ by letting each $y_i$ act by $c$ and $S_n$ act trivially.
Given a partition $\pi$ of $n =n_1 + \dots + n_k$ and $c_1, \dots, c_k \in \mathbb{C}$, we can consider the standard induced module [ ```$$ M^{\pi}_c = \mathrm{Ind} ( M^{n_1}_{c_1} \otimes \dots \otimes M^{n_k}_{c_k} ]
M^{n_k}_{c_k}) $$``` where the induction is from the subalgebra $T_{n_1} \otimes \dots \otimes T_{n_k} \subset T_n$ to $T_n$.

As far as I know, the simple quotients of these standard modules form a complete class for the simple representations of $T_n$. (I think this is a general fact about semidirect products of a finite group with a commutative algebra, together with the fact that representations of the symmetric group $S_n$ can be obtained by taking quotients induction of the trivial representation of a Young subgroup $S_{n_1} \times \dots \times S_{n_k}$.)

My question is how to represent this explicitly in terms of the simple objects in the semisimple category $Rep(S_n)$. First of all, the $S_n$-module $M^{\pi}_c$ can be described in a combinatorial manner in terms of the simple objects in $Rep(S_n)$ using the Kostka numbers. The action of the $y_i$'s on $M^{\pi}_c$ can be given in terms of a suitable morphism $y: \mathfrak{h} \otimes M^{\pi}_c \to M^{\pi}_c$, where $\mathfrak{h} \in Rep(S_n)$ is the regular representation.

The Pieri rule allows one to compute the decomposition in irreducibles (as parametrized by Young diagrams, of course) of $\mathfrak{h} \otimes M^{\pi}_c$, so we can view $y$ as a bunch of matrices based upon this decomposition (matrices w.r.t. the simple objects in $Rep(S_n)$, not as vector spaces).

Is there an approach to compute these matrices?

I am interested in this because it may allow a way to directly interpolate the construction of these modules to complex rank, via Pavel Etingof's program. (I believe one can interpolate the construction in another way, by reasoning more directly on the definition of the category $Rep(S_t)$ given by Deligne, but this seems to be useless as far as explicit computations--which might be helpful to study the "degeneracy phenomena" that Etingof has suggested might exist--are concerned.) In this case we have tensor categories $Rep(S_t)$ for $t$ not necessarily an integer, and while the interpretation in terms of vector spaces fails, the one in terms of Young diagrams does not.

1

# Explicit computation of induced modules of semidirect products with the symmetric group

I've gotten stuck in a project I have been working on, essentially on the following question.

One can obtain a 1-dimensional representation $M^n_c$ of the algebra $T_n := S_n \rtimes \mathbb{C}[y_1, \dots, y_n]$ by letting each $y_i$ act by $c$ and $S_n$ act trivially.
Given a partition $\pi$ of $n =n_1 + \dots + n_k$ and $c_1, \dots, c_k \in \mathbb{C}$, we can consider the standard induced module [ M^{\pi}_c = \mathrm{Ind} ( M^{n_1}_{c_1} \otimes \dots \otimes M^{n_k}_{c_k} ] where the induction is from the subalgebra $T_{n_1} \otimes \dots \otimes T_{n_k} \subset T_n$ to $T_n$.

As far as I know, the simple quotients of these standard modules form a complete class for the simple representations of $T_n$. (I think this is a general fact about semidirect products of a finite group with a commutative algebra, together with the fact that representations of the symmetric group $S_n$ can be obtained by taking quotients induction of the trivial representation of a Young subgroup $S_{n_1} \times \dots \times S_{n_k}$.)

My question is how to represent this explicitly in terms of the simple objects in the semisimple category $Rep(S_n)$. First of all, the $S_n$-module $M^{\pi}_c$ can be described in a combinatorial manner in terms of the simple objects in $Rep(S_n)$ using the Kostka numbers. The action of the $y_i$'s on $M^{\pi}_c$ can be given in terms of a suitable morphism $y: \mathfrak{h} \otimes M^{\pi}_c \to M^{\pi}_c$, where $\mathfrak{h} \in Rep(S_n)$ is the regular representation.

The Pieri rule allows one to compute the decomposition in irreducibles (as parametrized by Young diagrams, of course) of $\mathfrak{h} \otimes M^{\pi}_c$, so we can view $y$ as a bunch of matrices based upon this decomposition (matrices w.r.t. the simple objects in $Rep(S_n)$, not as vector spaces).

Is there an approach to compute these matrices?

I am interested in this because it may allow a way to directly interpolate the construction of these modules to complex rank, via Pavel Etingof's program. (I believe one can interpolate the construction in another way, by reasoning more directly on the definition of the category $Rep(S_t)$ given by Deligne, but this seems to be useless as far as explicit computations--which might be helpful to study the "degeneracy phenomena" that Etingof has suggested might exist--are concerned.) In this case we have tensor categories $Rep(S_t)$ for $t$ not necessarily an integer, and while the interpretation in terms of vector spaces fails, the one in terms of Young diagrams does not.
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