## Polynomials and Representations XXXIII

We are back to the convention $G = GL_n\mathbb{C}$ and $K = U_n.$ We wish to focus on irreducible polynomial representations of G.

The weak Peter-Weyl theorem gives:

$\displaystyle\mathcal{O}(K) \cong \bigoplus_{K-\text{irrep } V} \text{End}(V)^\vee = \bigoplus_{\text{rat. } G-\text{irrep } V} \text{End}(V)^\vee.$

Theorem. Restricting the RHS to only polynomial irreducible V gives us $\mathbb{C}[z_{ij}]_{1\le i, j\le n}$ on the LHS, where each polynomial $f$ in $z_{ij}$ restricts to a function $K \to \mathbb{C}.$

Proof

Since $z_{ij} \in \mathcal{O}(K)$ is a matrix coefficient, we have $\mathbb{C}[z_{ij}] \subseteq \mathcal{O}(K)$ and this is clearly a $G\times G$-submodule. Furthermore, as functions $K\to\mathbb{C}$, the $z_{ij}$ are algebraically independent over $\mathbb{C}$ by the main lemma here.

Since each $\text{End}(V)^\vee$ is irreducible as a $G\times G$-module, $\mathbb{C}[z_{ij}]$ corresponds to a direct sum $\oplus_S \text{End}(V)^\vee$ over some set S of G-irreps V. It remains to show that S is precisely the set of polynomial irreps.

Next, as $G\times G$-representations, we have a decomposition

$\displaystyle\mathbb{C}[z_{ij}] = \bigoplus_{d\ge 0} \mathbb{C}[z_{ij}]^{(d)}$

into homogeneous components; each is a finite-dimensional representations of $G\times G.$ By considering the action of $1\times G$, each component is a polynomial representation of G. Hence every irrep in S must be polynomial.

Conversely, if W is any polynomial irrep of G of dimension m, upon taking a basis the action of every $g\in G$ can be written as an $m\times m$ matrix with entries in $\mathbb{C}[z_{ij}].$ Hence $\mathbb{C}[z_{ij}]^m$ contains W; since W is irreducible, $\mathbb{C}[z_{ij}]$ contains W. ♦

## Decomposing By Degrees

Thus we have as isomorphism of $G\times G$-reps (*):

$\displaystyle\bigoplus_{\text{poly irrep } V \text{of }G} \text{End}(V)^\vee \cong \mathbb{C}[z_{ij}] = \bigoplus_{d\ge 0} \mathbb{C}[z_{ij}]^{(d)}.$

Definition. The degree of a polynomial irrep V is the unique $d\ge 0$ for which $\mathbb{C}[z_{ij}]^{(d)}$ contains it.

Let us pick one particular component $\mathbb{C}[z_{ij}]^{(d)}.$ We get a G-rep by taking the action of $1\times G.$ Hence its Laurent polynomial can be computed by considering the action of $1\times S$ on it. Since the space is spanned by monomials of degree d, we have:

Property 1. For a polynomial G-irrep V, we have $\deg V = \deg \psi_V.$

Recall that if G acts on V, then $G\times G$ acts on $\text{End}(V)$ via:

$(x,y) \in G\times G, f\in \text{End}(V) \ \mapsto \ (x,y)f = \rho_V(x)\circ f\circ \rho_V(y^{-1}).$

Hence, taking the dual gives:

Property 2. The action of $S\times S \subset G\times G$ on $\text{End}(V)^\vee$ gives the character

$\displaystyle \psi_V(x_1^{-1}, \ldots, x_n^{-1}) \psi_V(y_1, \ldots, y_n)$

The pair of diagonal matrices $(D(x_1, \ldots, x_n), D(y_1, \ldots, y_n)) \in S\times S$ takes $z_{ij} \mapsto x_i^{-1} z_{ij} y_j$. Hence, taking the basis of monomials of degree d, we have:

Property 3. The action of $S\times S\subset G\times G$ on $\mathbb{C}[z_{ij}]^{(d)}$ has character:

$\displaystyle \sum_{\substack{m_{11}, m_{12},\ldots, m_{nn} \ge 0\\ m_{11} + m_{12} + \ldots + m_{nn} = d}} \left( \prod_{i=1}^n \prod_{j=1}^n x_i^{-m_{ij}} y_j^{m_{ij}}\right).$

Finally, from lemma 3 here and property 1 above, we see that:

Property 4. For each d, the number of polynomial G-irreps of degree d is exactly the cardinality of $\{\lambda \vdash d: \lambda_1 \le n\}.$

## Main Theorem

Now we are ready to prove:

Theorem. A polynomial representation V of G of degree d is irreducible if and only if $\psi_V$ is a Schur polynomial in $x_1, \ldots, x_n$ of of degree d.

Proof

For each d, let $V_{d1}, \ldots, V_{de}$ be the polynomial irreps of degree d; let $p_{dj} := \psi_{V_{dj}},$ a homogeneous symmetric polynomial in $x_1, \ldots, x_n$ of degree d. By property 2, the character of $S\times S$ on $\mathbb{C}[z_{ij}]^{(d)}$ is:

$\displaystyle\sum_{j} p_{dj}(x_1^{-1}, \ldots, x_n^{-1}) p_{dj}(y_1, \ldots, y_n).$

By property 3 and (*), summing this over all d and j gives the power series:

$\displaystyle\sum_{m_{11}, \ldots, m_{nn}\ge 0} \left( \prod_{i=1}^n \prod_{j=1}^n x_i^{-m_{ij}} y_j^{m_{ij}}\right) = \prod_{i=1}^n \prod_{j=1}^n \sum_{m\ge 0} (x_i^{-1} y_j)^m = \prod_{1\le i, j\le n} \frac 1 {1 - x_i^{-1}y_j}.$

Finally by property 4, for each d, the number of $p_{dj}$ is exactly the size of $\{\lambda \vdash d : \lambda_1 \le n\}.$ Thus by the criterion for orthonormal basis proven (much) earlier, the $\{p_{dj}\}_j$ forms an orthonormal basis of $\Lambda_n^{(d)}.$ Hence, each $p_{dj}$ is, up to sign, a Schur polynomial of degree d. Since the coefficients of $p_{dj}$ are non-negative, they are the Schur polynomials. ♦

## Summary

We have a correspondence between:

which takes $V\mapsto \psi_V.$ Under this correspondence, for partition $\lambda = (\lambda_1, \ldots, \lambda_l)$,

\begin{aligned}\text{Sym}^{\lambda_1} \mathbb{C}^n \otimes \ldots \otimes \text{Sym}^{\lambda_l} \mathbb{C}^n &\leftrightarrow h_\lambda(x_1, x_2, \ldots, x_n), \\\text{Alt}^{\lambda_1} \mathbb{C}^n \otimes \ldots \otimes \text{Alt}^{\lambda_l} \mathbb{C}^n &\leftrightarrow e_\lambda(x_1, x_2, \ldots, x_n), \\ \text{poly. irrep } &\leftrightarrow s_\lambda(x_1, x_2, \ldots, x_n), \\ \otimes \text{ of reps } &\leftrightarrow \text{ multiplication of polynomials},\\ \text{Hom}_G(V, W) &\leftrightarrow \text{ Hall inner product}. \end{aligned}

Indeed, the first two correspondences are obvious; the third is what we just proved. The fourth is immediate from the definition of $\psi_V$. The final correspondence follows from the third one. Denote $V(\lambda)$ for the corresponding irrep of $GL_n\mathbb{C}$; we can now port over everything we know about symmetric polynomials, such as:

\displaystyle \begin{aligned} h_\lambda = \sum_{\mu\vdash d} K_{\mu\lambda} s_\mu \ &\implies\ \bigotimes_i \text{Sym}^{\lambda_i} \mathbb{C}^n = \bigoplus_{\mu\vdash d} V(\mu)^{\oplus K_{\mu\lambda}},\\ e_\lambda = \sum_{\mu\vdash d} K_{\overline\mu\lambda} s_\mu\ &\implies\ \bigotimes_i \text{Alt}^{\lambda_i} \mathbb{C}^n = \bigoplus_{\mu\vdash d} V(\mu)^{\oplus K_{\overline\mu\lambda}},\\ s_\lambda s_\mu = \sum_{\nu\vdash d} c_{\lambda\mu}^\nu s_\nu \ &\implies\ V(\lambda) \otimes V(\mu) \cong \bigoplus_{\nu\vdash d} V(\nu)^{\oplus c_{\lambda\mu}^\nu}.\end{aligned}

Setting $\lambda = (1,\ldots, 1)$ for $h_\lambda$ gives:

$\displaystyle(\mathbb{C}^n)^{\otimes d} \cong \bigoplus_{\mu\vdash d} V(\mu) ^{d_\mu}$

where $d_\mu$ is the number of SYT of shape $\mu.$

Unfortunately, the involution map $\omega : \Lambda_n \to \Lambda_n$ does not have a nice interpretation in our context. (No it does not take the polynomial irrep to its dual!)

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