Specht modules for symmetric group $S_{\infty}$ Specht modules of $S_n$, the symmetric group on n symbols is well-known.
Is there an analogue of these modules for $S_{\infty}$, the set of all permutations of $\mathbb N$?
Also, please share some references to learn the representation theory of $S_{\infty}$.
Thank you.
 A: There are at least two possible analogues here, depending on which features of Specht modules you consider to be most essential. However, I would like to emphasise that in both cases, they are representations of the group of all "finitary" permutations of $\mathbb{N}$ (permutations that fix all but finitely many elements), rather than the group of arbitrary permutations of $\mathbb{N}$.
This finitary version can be described as
$\varinjlim S_n$, with respect to the standard inclusions of symmetric groups (if $m \leq n$, $S_m$ is the subgroup of $S_n$ fixing all numbers larger than $m$). So it is closely tied to the finite symmetric groups, and in this answer, I will only consider this finitary version.
There is also the theory of Deligne categories (usually written $\underline{Rep}(S_t)$), which interpolate the Specht modules in a way that is similar to what is described in the first analogue below. Since the theory is more abstract (doesn't directly have modules or characters), I leave it out.

Analogue 1: Stable Representation Theory of Sam and Snowden.
Let's consider the simplest non-trivial Specht module $S^{(n-1,1)}$. If $\mathbb{C}^n$ is the permutation representation of $S_n$, it contains an invariant subspace $ \mathbb{C}(1,1,\ldots,1)$. A complement to this invariant subspace is given by mean-zero vectors: $\{(x_1, x_2, \ldots, x_n) \mid \sum_i x_i = 0\}$; the condition $\sum_i x_i = 0$ is preserved under permutation of the coordinates. This subspace is the Specht module $S^{(n-1,1)}$.
Notice that this construction is "uniform in $n$". So we could have replaced $\mathbb{C}^n$ by an infinite-dimensional vector space $\mathbb{C}^\infty$. This space no longer contains an invariant vector, because the all-ones vector $(1,1,\ldots)$ is no longer a finite linear combination of basis vectors. Nevertheless, there is a mean-zero subrepresentation defined just as before, and it is irreducible. The map $\mathbb{C}^\infty \to \mathbb{C}$ given by $(x_1, x_2, \ldots) \mapsto \sum x_i$ is a homomorphism having this mean-zero subrepresetation as its kernel. Thus the trivial representation is a quotient, but not a subrepresentation of the permutation representation. Here we encounter a point of difference from the finite theory; these representations are no longer semisimple.
Also instead of labelling this representation by the "partition" $(\infty -1, 1)$, we just remember all entries after the first; in this case it is just $(1)$. To construct more complicated Specht modules, we can instead look inside tensor powers $(\mathbb{C}^n)^{\otimes k}$; it turns out that the result will contain all Specht modules indexed by partitions with at most $k$ boxes below the top row. The construction again may be generalised to the case $n=\infty$, which according to our convention will give irreducible representations labelled by partitions of size at most $k$ (since we omit the first row which would contain the symbol $\infty$).
I have been a bit vague about how to find a given Specht module inside $(\mathbb{C}^n)^{\otimes k}$, but this essentially relies on the Schur-Weyl duality that holds between the symmetric group and the partition algebra. The details of this construction, as well as some basic properties of the category $Rep(S_\infty)$ in which these Specht modules naturally live, are described in the work of Sam and Snowden. See section 6 of the following paper:
Sam, Steven V.; Snowden, Andrew, Stability patterns in representation theory, Forum Math. Sigma 3, Paper No. e11, 108 p. (2015). ZBL1319.05146.
The category $Rep(S_\infty)$ is closed under the usual tensor product of representations (although it is far from being all representations of the abstract group $S_\infty$). If we write $V_\lambda$ for the the object generalising the Specht module $S^{(n-|\lambda|, \lambda)}$, then the tensor product multiplicities are the reduced Kronecker coefficients.
Now, notice that $S_\infty$ contains a subgroup $S_n \times H_n$, where the factor $S_n$ permutes the numbers $\{1,2, \ldots, n\} \subseteq n$, and the factor $H_n$ is abstractly isomorphic to $S_\infty$, but permutes the numbers $\{n+1, n+2, \ldots\}$. If $V$ is in $Rep(S_\infty)$, then the $H_n$ invariants $V^{H_n}$ retain an action of $S_n$ (because $S_n$ commutes with the $H_n$-action). Thus we get a "specialisation" functor $F_n$ from $Rep(S_\infty)$ to $Rep(S_n)$ for any $n$. This turns out to be a monoidal functor.
A consequence of defining things in this way is that $F_n(V_\lambda) = S^{(n-|\lambda|, \lambda)}$, provided that $(n-|\lambda|, \lambda)$ is a partition (i.e. that $n-|\lambda| \geq \lambda_1$). If that is not the case ($n$ is too small), then instead $F_n(V_\lambda) = 0$. One may wish to compute the derived functors of $F_n$, and this can be done by constructing a suitable injective resolution in $Rep(S_\infty)$. One way to do this is using an analogue of the BGG resolution; see Theorem 2.3.1 of
Sam, Steven V.; Snowden, Andrew, GL-equivariant modules over polynomial rings in infinitely many variables, Trans. Am. Math. Soc. 368, No. 2, 1097-1158 (2016). ZBL1436.13012.
We may view the symmetric group as the subgroup of the general linear group consisting of permutation matrices. It turns out that objects obtained by restricting representations of $GL_\infty$ to $S_\infty$ are injective in $Rep(S_\infty)$. So a different strategy is to understand how representations of general linear groups restrict to symmetric groups. Although this is not very well understood (it is an open problem to give a combinatorial formula known for the restriction multiplicities), one can use this approach to construct such an injective resolution; see
Ryba, Christopher, Littlewood complexes for symmetric groups,  ZBL07383251.
One upshot of this is as follows. There is a good notion of the "character" of irreducible representations of general linear groups - they are Schur functions, which form a basis of the ring of symmetric functions, $\Lambda$. This connection between general linear groups and symmetric groups lets us identify the Grothendieck ring of $GL_\infty$ (i.e. $\Lambda$) with the Grothendieck ring of $Rep(S_\infty)$. Thus we obtain "characters" for the $V_\lambda$, which are symmetric functions $\tilde{s}_\lambda$ introduced independently in the following two papers:
Orellana, Rosa; Zabrocki, Mike, Symmetric group characters as symmetric functions, Adv. Math. 390, Article ID 107943, 34 p. (2021). ZBL1473.05309.
Assaf, Sami H.; Speyer, David E., Specht modules decompose as alternating sums of restrictions of Schur modules, Proc. Am. Math. Soc. 148, No. 3, 1015-1029 (2020). ZBL1455.20007.
The "character" of $V_\lambda$ allows us to compute the characters of the usual Specht modules $S^{(n-|\lambda|, \lambda)}$, strengthening the claim that $V_\lambda$ is the correct notion of Specht module for $S_\infty$. (Although this is a bit subtle - to get a character value, we must evaluate a symmetric function at the eigenvalues of a permutation matrix, just as one would evaluate a Schur function at the eigenvalues of an element of $GL_n$.)

Analogue 2: Edrei-Thoma theorem
In the previous discussion, I mentioned that $Rep(S_\infty)$ is quite far away from being all representations of $S_\infty$. For example, $S_\infty$ has a sign representation, which does not appear in that category. We will discuss a theory that includes the sign representations (and many others not incuded in $Rep(S_\infty)$).
In this setting we will eschew Specht modules in favour of their characters. What is the right notion of a character of (possibly infinite-dimensional) representation of an infinite group? Well, suppose that $\chi : G \to \mathbb{C}$ is the character of a representation of a finite group. We normalise this character: let $\phi: G \to \mathbb{C}$ be defined by $\phi(g) = \chi(g)/\chi(1)$. This has the following properties:

*

*$\phi(\mathrm{Id}) = 1$

*$\phi(g_1 g_2) = \phi(g_2 g_1)$

*For any finite sequence of group elements $g_i$, the matrix $\phi(g_1g_2^{-1})$ is Hermitian and positive semidefinite

Some comments are in order. The first condition is trivial, but hides the essential detail that characters in the usual sense are poorly behaved on infinite dimensional representations (e.g. the trace of the identity is infinite). So instead of looking for genuine characters, we will look for such normalised characters. The second condition says that our function is a class function, as any character would be. The third condition has two parts. Being Hermitian requires $\chi(g_1g_2^{-1}) = \overline{\chi(g_2g_1^{-1})}$ , which holds for finite groups because the eigenvalues of the representing matrices are roots of unity (more generally this works for compact groups). The definiteness condition for such sequences follows from the particular case where $g_1, g_2, \ldots, g_l$ enumerates all the elements in the group (by taking a submatrix of this matrix). However, in order to avoid dealing with infinite matricess when the group is infinite, we stick to finite submatrices.
Finally let me add that the normalised character corresponding to $\chi_1 \oplus \chi_2$ is
$\frac{\chi_1(g) + \chi_2(g)}{\chi_1(\mathrm{Id}) + \chi_2(\mathrm{Id})} = \frac{\chi_1(g)}{\chi_1(\mathrm{Id})} \cdot \frac{\chi_1(\mathrm{Id})}{\chi_1(\mathrm{Id}) + \chi_2(\mathrm{Id})} + \frac{\chi_1(g)}{\chi_1(\mathrm{Id})} \cdot \frac{\chi_2(\mathrm{Id})}{\chi_1(\mathrm{Id}) + \chi_2(\mathrm{Id})}$
which is in particular a convex combination of the normalised characters for $\chi_1$ and $\chi_2$. So if our representation was reducible, it is a non-trivial convex combination of other normalised characters. Thus we arrive at our definition of "irreducible character of $S_\infty$" - it is a function on the group obeying the three properties stated above, which cannot be written as a non-trivial convex-combination of other such functions.
For $S_\infty$, such functions have been classified, and this is the Edrei-Thoma theorem. Since this post is already very long, I will just summarise the result. The functions in question are parametrised by pairs of infinite sequences of non-negative real numbers $(\alpha_i, \beta_i)$, where $\alpha_1 \geq \alpha_2 \geq \cdots$ and $\beta_1 \geq \beta_2 \geq \cdots$, and $\sum_i \alpha_i + \sum_j \beta_j \leq 1$. We let $\gamma = 1 - \sum_i \alpha_i - \sum_j \beta_j$ (it is a real number between 0 and 1). This odd-looking parametrisation can be interpreted in terms of partitions as follows.
Let us consider a sequence of normalised characters, viewed as functions on successively larger symmetric groups. Provided some kind of suitable convergence holds, we will get a normalised character on $S_\infty$. We take these normalised characters to correspond to progressively larger partitions $\lambda^{(n)}$ (indexing Specht modules). Then $\lambda_i^{(n)} / |\lambda^{(n)}|$ describes the fraction of the partition that lies in the $i$-th row. Similarly $(\lambda^{(n)})_j^\prime / |\lambda^{(n)}|$ describes the fraction of the partition that lies in the $j$-th column. In the limit, these quantities converge to the parameters $\alpha_i$ and $\beta_j$, and the remainder, $\gamma$ describes the fraction that is left over and distributed in the "bulk" of the diagram. There are explicit formulae for the normalised character in terms of the $\alpha_i$ and $\beta_j$. For more detail, one can consult the following references (which don't seem to appear in the citation engine):
Okounkov - On the representations of the infinite symmetric group
Borodin and Olshanski - Representations of the Infinite Symmetric Group
So for this definition of irreducible (normalised) character, we get uncountably many. While this approach may be broader than the first, what we get is in some ways quite different from the classical Specht modules.
