On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras

TL;DR

This paper investigates the asymptotic behavior of irreducible representation multiplicities in large tensor products of simple Lie algebra representations, revealing universal patterns and dependencies on parameters.

Contribution

It generalizes previous results by deriving the asymptotic distribution of irreducibles under Plancherel and character measures for large tensor products.

Findings

Asymptotic distribution of irreducible components under Plancherel measure

Universal asymptotic behavior of character measure near Plancherel measure

Dependence of asymptotics on parameter degeneracy

Abstract

In this paper we study the asymptotic of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of Biane and Tate and Zelditch. We also derive the asymptotic of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.

Equations

$$\underset{k=1}{\overset{m}{⨂}}{V}_k^{\otimes N_k}\cong \underset{\lambda }{⨁}{W}_{\lambda }(\{V_k\},\{N_k\})\otimes V_{\lambda },$$

$$W_{\lambda }(\{V_k\},\{N_k\})\simeq {\mathrm{Hom}}_{\mathfrak{g}}(\underset{k=1}{\overset{m}{⨂}}{V}_k^{\otimes N_k},V_{\lambda }).$$

$$\prod _k{\chi }_{{\nu }_k}(e^t{)}^{N_k}=\sum _{\lambda }{m}_{\lambda }(\{V_k\},\{N_k\}){\chi }_{\lambda }(e^t).$$

$$p_{\lambda }^{(N)}(t)=\frac{m_{\lambda }(\{V_k\},\{N_k\}){\chi }_{\lambda }(e^t)}{{\prod }_k{\chi }_{{\nu }_k}(e^t{)}^{N_k}}$$

$$f(\tau ,t)=\sum _k{\tau }_k\ln ({\chi }_{{\nu }_k}(e^t)).$$

$$S(\tau ,\xi )=\underset{y}{\min }\left(f(\tau ,y)-(y,\xi )\right)=f(\tau ,x)-(x,\xi ),$$

$$\frac{\partial }{\partial x_a}f(\tau ,x)=\sum _b{B}_{ab}{\xi }_b.$$

$$K_{ac}=-\frac{{\partial }^{2}S(\tau ,\xi )}{\partial {\xi }_a\partial {\xi }_b},$$

$$K_{ac}=\sum _{b,d}{B}_{ad}(D^{-1}{)}_{db}{B}_{bc},$$

$$D_{ab}=\frac{{\partial }^2}{\partial y_a\partial y_b}f(\tau ,y){|}_{y=x},$$

$$m_{\lambda }(\{V_k\},\{N_k\})={ϵ}^{\frac{r}{2}}\frac{\sqrt{detK}}{(2\pi {)}^{\frac{r}{2}}}\Delta (x)e^{-(\rho ,x)}{e}^{\frac{1}{ϵ}S(\tau ,\xi )}(1+O(ϵ))$$

$$\Delta (x)=\prod _{\alpha \in {\Delta }_{+}}(e^{\frac{(x,\alpha )}{2}}-e^{-\frac{(x,\alpha )}{2}})$$

$$p_{\lambda }^{(N)}(t)={ϵ}^{\frac{r}{2}}\frac{\sqrt{detK}}{(2\pi {)}^{\frac{r}{2}}}\frac{\Delta (x)}{\Delta (t)}{e}^{-(\rho ,x-t)}{e}^{\frac{1}{ϵ}\tilde{S}(\tau ,\xi )}(1+O(ϵ))$$

$$p(a,t)=\frac{\sqrt{detK}}{(2\pi {)}^{\frac{r}{2}}}{e}^{-\frac{1}{2}{\sum }_{bc}{a}_b{K}_{bc}{a}_c}.$$

$$\sum _{\lambda \in D}h(\sqrt{{ϵ}_n}(\lambda -\frac{\eta }{{ϵ}_n}))p_{\lambda }^{(N)}(t)\to {\int }_{{\mathbb{R}}^r}h(a)p(a,t)d{a}_1\dots d{a}_r,$$

$$p_{\lambda }^{(N)}(0)=\frac{m_{\lambda }(\{V_k\},\{N_k\})dim(V_{\lambda })}{{\prod }_{k}dim(V_{{\nu }_k}{)}^{N_k}}.$$

$$p(a)=\frac{\sqrt{detB}}{(2\pi {)}^{\frac{r}{2}}}\prod _{\alpha >0}\frac{(a,\alpha {)}^2}{(\rho ,\alpha )}{e}^{-\frac{1}{2}{\sum }_{b,c}{a}_b{B}_{bc}{a}_c},$$

$$p(b,u)=\frac{\sqrt{\det B}}{(2\pi {)}^{\frac{r}{2}}}\frac{{\prod }_{\alpha \in {\Delta }_{+}}(b,\alpha )}{{\prod }_{\alpha \in {\Delta }_{+}}(u,\alpha )}\sum _{w\in W}(-1{)}^w{e}^{(b,w(u))}{e}^{-\frac{(b,b)}{2}-\frac{(u,u)}{2}}$$

$$W\simeq \oplus V_i^{\oplus m_i}$$

$$p_i=\frac{dim(V_i)m_i}{dim(W)}.$$

$$\rho =\frac{I}{dim(W)},$$

$$\rho =\frac{e^{-\frac{H}{T}}}{tr(e^{-\frac{H}{T}})}$$

$$m_{\lambda }^N=e^{\frac{1}{ϵ}S(\tau ,\xi )}\mu (\tau ,\xi )(1+O(ϵ)).$$

$$\prod _k{\chi }_{{\nu }_k}(e^t{)}^{N_k}=\sum _{\lambda \in D}{m}_{\lambda }(\{V_k\},\{N_k\}){\chi }_{\lambda }(e^t).$$

$$e^{\frac{f(\tau ,t)}{ϵ}}=\sum _{\lambda }{m}_{\lambda }^N{\chi }_{\lambda }(e^t)\simeq {ϵ}^{-r}{\int }_{D_0}{e}^{\frac{1}{ϵ}S(\tau ,\xi )}\mu (\tau ,\xi ){\chi }_{\frac{\xi }{ϵ}}(e^t)d\xi ,$$

$${\chi }_{\lambda }(e^t)=\frac{{\sum }_w{e}^{(w(\lambda +\rho ),t)}(-1{)}^{l(w)}}{\Delta (t)},$$

$${\chi }_{\frac{\xi }{ϵ}}(e^t)=e^{\frac{1}{ϵ}(\xi ,t)}\frac{e^{(\rho ,t)}}{\Delta (t)}(1+o(1)).$$

$$e^{\frac{f(\tau ,t)}{ϵ}}\simeq {ϵ}^{-r}{\int }_{D_0}{e}^{\frac{1}{ϵ}S(\tau ,\xi )}\mu (\tau ,\xi ){\chi }_{\frac{\xi }{ϵ}}(x)d\xi \simeq {ϵ}^{-r}{\int }_{{\mathbb{R}}_{\ge 0}^r}{e}^{\frac{1}{ϵ}(S(\tau ,\xi )+(\xi ,t))}\mu (\tau ,\xi )\frac{e^{(\rho ,t)}}{\Delta (t)}d\xi .$$

$$e^{\frac{f(\tau ,t)}{ϵ}}=e^{\frac{1}{ϵ}(S(\tau ,\eta )+(t,\eta ))}\mu (\tau ,\eta ){ϵ}^{-r}\frac{e^{(\rho ,t)}}{\Delta (t)}{ϵ}^{\frac{r}{2}}{\int }_{{\mathbb{R}}^r}{e}^{\frac{1}{2}(a,S^{(2)}a)}{d}^{r}a$$

$$S(\tau ,\xi )=\underset{y}{\min }f(\tau ,y)-(y,\xi ))=f(\tau ,x)-(x,\xi ),$$

Full text

1. Introduction

1.1.

The study of the statistics of irreducible components in ‘‘large" natural representations goes back to works [12][17][18]. An example of such large representation is the left regular representation of the symmetric group $S_{N}$ for large $N$. The Plancherel measure provides a natural probability measure. The statistics of irreducible components in the left regular representation of $S_{N}$ with respect to the Plancherel measure was exactly the focus of [12][17][18],[5], see also [6][13] and references therein.

The Schur-Weyl duality identifies multiplicities of irreducible components of the $N$-th tensor power of the vector representation of $sl_{n+1}$ with dimensions of irreducible representations of $S_{N}$. The statistics of irreducible $sl_{n+1}$-components in this tensor products with respect to the Plancherel measure111If $W$ is a finite dimensional representation of a simple Lie algebra and $W\simeq\oplus_{i}V_{i}^{\oplus m_{i}}$ is its decomposition into irreducibles, then $p_{i}=\frac{\dim V_{i}m_{i}}{\dim W}$ is a probability measure on the set of irreducible components of $W$. We call it the Plancherel measure by the analogy with the Plancherel measure on the left regular representation of a finite group, or compact Lie group. (8) for large $N$ was studied by Kerov in [9]. Kerov [9] discovered that as $N\to\infty$ the discrete measure (8) on the space of highest weights $\mathbb{R}^{n}$ of $sl_{n+1}$ converges weakly to the measure on the main Weyl chamber given by the radial part of the natural invariant measure on $sl_{n+1}^{*}$. Kerov’s proof was based upon the hook formula for dimensions of irreducible representations of $S_{N}$.

Among more recent results involving similar computations are papers [1],[2] where similar asymptotics were computed for Lie superalgebras $gl(n|m)$, the paper [14], where the asymptotic of multiplicities of irreducible $so(2n+1)$-modules in the $N$-th tensor powers of the spinor representation was studied in the limit $N\to\infty$ and [6]. In [4] the asymptotic of multiplicities of irreducible subrepresentations was computed for $V^{\otimes N}$ when $N\to\infty$. These results were extended and strengthened in [16], see also [15]. Problems of this type are known as asymptotic representation theory. A survey of this direction in representation theory can be found in the books [10], [19].

In this paper we derive the asymptotic formula for multiplicities of irreducible representations in tensor powers of finite dimensional representations of simple Lie algebras when the number of factors tends to infinity. The formula generalizes slightly the one obtained in [16],[4]. Here we derive it using a different method, somewhat heuristically. The complete proof is a straightforward extension of results from [16]. The large deviation rate function was also computed in [8] and the convergence of character measures was studied in [KW].

1.2.

Let $\mathfrak{g}$ be a simple Lie algebra, $V_{i},\;\;i=1\dots m$ be its finite dimensional representations and $N_{k}\geq 0$ be integers. Any finite dimensional representation of a simple Lie algebra is completely reducible and therefore:

[TABLE]

where the sum is taken over irreducible components of the tensor product, $V_{\lambda}$ is the irreducible finite dimensional $\mathfrak{g}$-module with the highest weight $\lambda$ and $W_{\lambda}(\{V_{k}\},\{N_{k}\})$ is the "space of multiplicities":

[TABLE]

Its dimension $m_{\lambda}(\{V_{k}\},\{N_{k}\})$ is the multiplicity of $V_{\lambda}$ in the tensor product and we have isomorphism of vector spaces.

Starting from here we assume $V_{k}=V_{\nu_{k}}$, otherwise $V_{k}\simeq\bigoplus V_{\nu}^{\oplus m_{\nu,k}}$. We will use notations $\chi_{\lambda}(\mathfrak{g})=tr_{V_{\lambda}(\mathfrak{g})}$ for characters of irreducible representations.

Choose a Borel subalgebra $\bf\in\mathfrak{g}$ and let $\Delta_{+}$ be corresponding positive roots, $\mathfrak{h}$ be the corresponding Cartan subalgebra and $\alpha_{1},\dots,\alpha_{r}$ be enumerated fundamental roots. Here $r=rank(\mathfrak{g})=dim(\mathfrak{h})$ is the rank of the Lie algebra $\mathfrak{g}$. Let $\mathfrak{g}_{\mathbb{R}}$ be the split real form of $\mathfrak{g}$ and $\mathfrak{h}_{\mathbb{R}}$ be its Cartan subalgebra. When $g=e^{t},\;\;t\in\subset\mathfrak{h}_{\mathbb{R}}\subset\mathfrak{g}_{\mathbb{R}}$ the characters $\chi_{\nu_{k}}(e^{t})$ and $\chi_{\lambda}(e^{t})$ are positive and as a consequence of the tensor product decomposition we have the identity

[TABLE]

Therefore

[TABLE]

is a natural probability measure on irreducible components of tensor product: $p_{\lambda}^{(N)}(t)\geq 0$ and $\sum_{\lambda}p_{\lambda}^{(N)}(t)=1$. We will call it the character measure.

We will also assume222This is a convenience assumption, the irreducible characters restricted to the Cartan subgroup are invariant with respect to the action of the Weyl group, so we can always choose a representative of the orbit of $W$ through $t$ which is positive. that components of $t$ in the basis of simple roots are nonnegative, i.e. $t\geq 0$.

In this paper we study the asymptotical behavior of multiplicities $m_{\lambda}$ in the limit when $N_{k}\to\infty$ and $\lambda\to\infty$ such that $N_{k}=\tau_{k}/\epsilon$ and $\lambda=\xi/\epsilon,\;\;\epsilon\to 0$, where $\tau_{k}\in\mathbb{R}_{\geq 0}$ and $\xi\in\mathfrak{h}^{*}_{\geq 0}$. As a consequence we will describe the large deviation type asymptotic of the probability measure $\{p_{\lambda}\}$.

1.3.

To state main results of the paper we need a few definitions. First, define

[TABLE]

It is a strictly convex function of $t$. Let $S(\tau,\xi)$ be the Legendre transform of $f(\tau,y)$

[TABLE]

where $(y,\xi)$ is the Killing form. In the basis of simple roots $\xi=\sum_{a}{\xi_{a}\alpha_{a}},y=\sum_{a}y_{a}\alpha_{a}$ and $(y,\xi)=\sum_{ab}y_{a}B_{ab}\xi_{b}$, where $B_{ab}=d_{a}C_{ab}=(\alpha_{a},\alpha_{b})$ is the symmetrized Cartan matrix and $x$ is the critical point where the minimum is achieved. The critical point $x$ is the unique solution to the equation:

[TABLE]

Define the matrix $K$ as

[TABLE]

we will see that

[TABLE]

where

[TABLE]

where $x$ is as above.

Recall that $y\in\mathfrak{h}$ is called regular if the stabilizer of $y$ in the Weyl group $W$ acting on $\mathfrak{h}$ is trivial. The following theorem extends the result of [4],[16] where it was proven for $m=1$.

Theorem 1**.**

If $\xi=\epsilon\lambda$ remain regular as $\epsilon\to 0$ the asymptotic of the multiplicity of $V_{\lambda}$ in (1) has the following form

[TABLE]

Here $x\in\mathfrak{h}$ is the Legendre image of $\xi\in\mathfrak{h}^{*}$, the functions $S$ and the matrix $K$ are as above and $\Delta(x)$ is the denominator in the Weyl formula for characters:

[TABLE]

Note that the asymptotic is different when $\xi$ is not regular. In this paper we give a heuristic argument why the theorem holds. A rigorous proof, based on the application of the steepest descent method to the integral of characters can be found in [16] for $m=1$ and for $m>1$ it is completely parallel.

The next theorem is the description of the statistics of irreducible components with respect to the character measure (2).

Assume $\xi=\epsilon\lambda$ remain regular as $\epsilon\to 0$ and $t$ is regular. Taking into account the asymptotic of the multiplicity and the asymptotic of the charachter $\operatorname{ch}_{\lambda}(e^{t})$ in this limit, we obtain the following asymptotic of the probability $p_{\lambda}^{(N)}(t)$ as $\epsilon\to 0$ is:

[TABLE]

where $\widetilde{S}(\tau,\xi)=S(\tau,\xi)-f(\tau,t)+(t,\xi)$. The exponent has maximum at $\eta$ which is the Legendre image of $t$ and $\widetilde{S}(\tau,\eta)=0$. Computing this expression in a neighborhood of the critical point $\eta$ we derive the following statement.

Theorem 2**.**

If $t$ is regular, the asymptotic character probability distribution is localized at point $\eta$ with a Gaussian distribution around this point. If we rescale random variable $\xi=\sum_{a}\xi_{a}\alpha_{a}$ near the critical point $\eta$ as $\xi_{l}=\eta_{l}+\sqrt{\epsilon}a_{l}$, then in the limit $\epsilon\to 0$ the random variable $a\in\mathfrak{h}^{*}_{\mathbb{R}}\simeq\mathbb{R}^{r}$333Here we use the basis of simple roots. is distributed with the probability measure $p(a)da_{1}\dots da_{n}$ where $da$ is the Euclidean measure on $\mathbb{R}^{r}$ and

[TABLE]

In other words, the probability measure $\{p_{\lambda}\}$ weakly converges to the probability measure $p(a)da$

Let $\Lambda$ be the weight lattice of the Lie algebra $\mathfrak{g}$ and $\epsilon_{n}$ be a positive sequence such that $\epsilon_{n}\to 0$ when $n\to\infty$. Define $\Lambda_{\epsilon_{n}}\subset\mathbb{R}^{r}$ as the image of $\Lambda$ with respect to the embedding $\lambda=\sum_{i}\lambda_{i}\omega_{i}\mapsto\sum_{i}\epsilon_{n}\lambda_{i}\omega_{i}\in\mathbb{R}^{r}$. Define the subset $D\subset\Lambda$ as the set of weights that appear in the decomposition of the tensor product (1) and the subset $D_{\epsilon_{n}}\subset\Lambda_{\epsilon_{n}}\subset\mathbb{R}^{r}$. Note that elements of $D$ have the form $\sum_{k}N_{k}\nu_{k}-\sum_{b=1}^{r}a_{b}\alpha_{b}$ where $a_{b}\geq 0$ are integers and $\alpha_{a}$ are simple roots. The weak convergence of measures in the context of theorem 2 means the following. For any bounded continuous function $h$ on $\mathbb{R}^{r}$, a sequence $\epsilon_{n}\to 0$, and the condition $\tau_{k}=\epsilon_{n}N_{k}$ are fixed for all $k$ in the tensor product

[TABLE]

The extreme non regular case is when $t=0$. In this case the probability distribution is given by

[TABLE]

Theorem 3**.**

As $\epsilon\to 0$ the Plancherel measure (8) weakly converges to the probability measure $p(a)da$ on $\mathfrak{h}^{*}_{\geq 0}\simeq\mathbb{R}^{r}$ where $da$ is the Euclidean measure on $\mathbb{R}^{r}$ and the density function is

[TABLE]

where random variables $\lambda$ and $a$ are related as $\lambda=\sqrt{\frac{1}{x\epsilon}}a$ and $x=\frac{1}{dim(\mathfrak{g})}\sum_{k=1}^{m}\tau_{k}c_{2}(\nu_{k})$. Here $c_{2}(\nu)$ is the value of the Casimir element on the irreducible representation $V_{\nu}$.

This result generalizes slightly convergence results for $m=1$ from [4][16].

Now consider again the character measure on irreducible components of the tensor product but assume that $t=\sqrt{\frac{\epsilon}{x}}u$ as $\epsilon\to 0$. In this case $p_{\lambda}^{(N)}(t)$ converges to a natural deformation of the measure (9).

Theorem 4**.**

As $\epsilon\to 0$, $t=\sqrt{\frac{\epsilon}{x}}u$ and $u$ is fixed, the character measure (2) weakly converges to the measure $p(b,u)db_{1}\dots db_{r}$ on $\mathbb{R}^{r}_{\geq 0}$ with density

[TABLE]

Here random variables $\lambda$ and $b$ as related as $\lambda=\sqrt{\frac{1}{x\epsilon}}b$.

1.4.

The paper is organized as follows. In section 3 we derive the asymptotic of the multiplicity function from the character identities. In section 4 we derive the asymptotic distribution with respect to the character measure (2), assuming that $t$ is regular. We also show there that the trivial representation is the most probable with respect to the Plancherel measure and derive the asymptotic statistics around it. Section 5 contains an outline of the derivation of the asymptotic of the multiplicity function using the hook formula for $N$-th tensor power of the vector representation of $sl_{n+1}$ and the corresponding asymptotic distribution. In section 6 we compute the same asymptotic from theorem 1. We derive intermediate asymptotic and outline the proof of Theorem 3 in section 7. Nonlinear PDE’s for the rate function are derived in section 8. In the conclusion we outline some perspectives.

1.5. Acknowledgments

We thank A. Nazarov and V. Serganova for stimulating discussions. We are grateful for E. Feigin, V. Gorin, M. Walter and S. Zelditch for important remarks and for pointing us to recent works on the subject. This work was supported by RSF-18-11-00-297. The work of NR was partly supported by the grant NSF DMS-1601947.

2. Plancherel and character measures as density matrices

Let $W$ be a finite dimensional representation of a simple Lie algebra $\mathfrak{g}$. Let

[TABLE]

be the decomposition of $W$ into irreducibles where $m_{i}$ is the multiplicity of $V_{i}$. By analogy with the usual Plancherel measure for left regular representations we will say that the Plancherel measure on $W$ is the probability measure on the set of irreducible components of $W$ with the probability of $V_{i}$ being

[TABLE]

This probability measure can be regarded as a quantum probabilistic state on $End(W)$ which assigns to an operator $A:W\to W$ its expectation value $tr(A)/dim(W)$. In other words, this is the state with the density matrix

[TABLE]

where $I:W\to W$ is the identity operator acting in $W$. This density matrix can be interpreted as the quantum equilibrium state

[TABLE]

with zero Hamiltonian, $H=0$, i.e. the topological quantum mechanics with the space of states $W$.

The character measure (2) can be interpreted in a similar way. It corresponds to the Hamiltonian being semisimple element of $\mathfrak{g}$, i.e. by a conjugation such an $H$ can be brought to an element of the Cartan subalgebra. In notations used in the introduction $t=-H/T$.

3. Derivation of the asymptotic of multiplicities

In this section we give arguments justifying conjecture 1 for the asymptotic of the multiplicity function for tensor product of representations and show how to obtain exponential and sub exponential terms of this asymptotic from their characters. From now on we assume that $\lambda$ is regular.

3.1. Large deviation rate function $S(\tau,\xi)$

We expect444This is proven in [16] for $m=1$. The proof is based on the application of the steepest descent method and on the orthogonality of characters of irreducible representations of corresponding compact Lie group. For $m>1$ the proof is completely similar. that in the limit $N=\tau/\epsilon$ and $\lambda=\xi/\epsilon,\;\;\epsilon\to 0$ with $\nu_{k}$ and $x$ fixed, the multiplicity function $m_{\lambda}^{N}$ has the following asymptotic:

[TABLE]

Here $S(\tau,\xi)$ is the large deviation rate function and $\mu(\tau,\xi)$ is a function of $\tau,\xi$ which is proportional to a power of $\epsilon$. We will now derive explicit expressions for these functions via character identities. Note that here $\lambda$ is in the main Weyl chamber and therefore $\xi\in\mathfrak{h}^{*}_{\geq 0}$ where $\geq 0$ means $\xi=\sum_{a=1}^{r}\xi_{a}\alpha_{a}$ has nonnegative coordinates $\xi_{a}$.

From the tensor product decomposition we have:

[TABLE]

where $\chi_{\lambda}(e^{t})=tr_{V_{\lambda}}(e^{t})$ is the character of module $V_{\lambda}$ evaluated at $e^{t}\in G$ and $D\subset\Lambda$ is as in (7).

Now let us substitute (11) into (12) and replace the summation over $\lambda$ by an integral over $\xi$:

[TABLE]

Here $r$ is the rank of Lie algebra $\mathfrak{g}$, $D_{0}\subset\mathbb{R}^{r}$ is the limiting set of $D_{\epsilon}$ as $\epsilon\to 0$. $D_{0}$ is a convex polytope in the positive chamber $\mathbb{R}^{r}_{\geq 0}$ with $\sum_{k}\tau_{k}\nu_{k}$ being the extreme point which is farthest from the origin. The measure $d\xi$ is the Euclidean measure $d\xi_{1}\cdots d\xi_{r}$.

Let us use the Weyl character formula

[TABLE]

where $\rho=\frac{1}{2}\sum_{\alpha\in\Delta^{+}}\alpha,$ and $l(w)$ is the length of $w$ (in terms of simple reflections). When $t$ is regular and is in the positive Weyl chamber and $\lambda=\xi/\epsilon$ where $\xi$ is fixed and is also in the positive Weyl chamber, and $\epsilon\to 0$, the term with $w=e$ gives the leading asymptotic and we have:

[TABLE]

From now on we will identify $\mathfrak{h}^{*}_{\geq 0}$ with $\mathbb{R}^{r}_{\geq 0}$ using the basis of simple roots.

Substituting (14) into (13) we obtain:

[TABLE]

where $d\xi=d\xi_{1}\dots d\xi_{r}$.

Assume that $S(\tau,\xi)$ is strictly concave in $\xi$. This implies that $S(\tau,\xi)+(\xi,t)$ has unique maximum at some point $\eta$. Then the leading contribution to the integral comes from the vicinity of $\eta$. It is given by the Gaussian integral over $a=(\xi-\eta)/\sqrt{\epsilon}$ and determined by the second Taylor coefficient of $S(\tau,\xi)$:

[TABLE]

where $S^{(2)}_{ab}=\frac{\partial^{2}}{\partial\xi_{a}\partial\xi_{b}}S(\tau,\xi)|_{\eta}$.

Comparing most singular terms in this equation we can identify $S(\tau,\xi)$ with the Legendre transform of $f(\tau,y)$ in $y$:

[TABLE]

Because the characters are strictly convex, the function $f(\tau,x)$ is strictly convex in $x$. This implies that $S(\tau,\xi)$ is strictly concave in $x$, which agrees with the assumption made earlier. Note that for the inverse Legendre transform we have

[TABLE]

where $x$ and $\eta$ are related as

[TABLE]

[TABLE]

Here $B_{ab}$ is the symmetrized Cartan matrix and $(\eta,\xi)=\sum_{a}\eta_{a}B_{ab}\xi_{b}$.

Since $f(\tau,x))$ is a convex function of $x$ with minimum at $x=0$, the function $S(\tau,\xi)$ defined in (17) is a concave function of $\xi$ with the unique maximum at $\xi=0$.

3.2. Subexponential terms

Now after deriving most singular exponential factors, the equation (16) is reduced to

[TABLE]

Taking into account that the Gaussian integral can be computed as

[TABLE]

the equation(21) gives:

[TABLE]

where $K_{ab}=-\frac{\partial^{2}S}{\partial\xi_{a}\partial\xi_{b}}(\xi)|_{\xi=\eta}$ and $\xi_{0}$ is the Legendre image of $x$.

Therefore, we derived the unknown function $\mu(\tau,\xi)$ as

[TABLE]

where $\eta$ and $x$ are related as in (19).

To calculate partial $K$ explicitly we will use (20) and (19). Differentiating (20) with respect to $x_{d}$ we get

[TABLE]

then

[TABLE]

To obtain $\frac{\partial x_{d}}{\partial\eta_{c}}$ we will differentiate (19) with respect to $\eta_{c}$:

[TABLE]

Here $x$ is regarded as the Legendre image of $\eta$. Denote

[TABLE]

then

[TABLE]

and therefore

[TABLE]

Finally, we derive the formula for $K$ in terms of $f$:

[TABLE]

4. Probability distributions

Here we will derive probability distributions for two cases, for the character measure and for the Plancherel measure, i.e. we will outline proofs of Theorems 2 and 3.

4.1. Character distribution.

Here we outline the proof of Theorem 2. The function

[TABLE]

in the formula (5) is strictly concave in $\xi$ and therefore it has unique maximum at some point $\eta$, which is the Legendre image of $t$ with respect to the function $S(\tau,\xi)$. Because $S(\tau,\xi)$ is invariant with respect to the action of the Weyl group, the point $\eta$ is regular if $t$ is regular. In the vicinity of $\eta$ we have

[TABLE]

where $a_{a}=(\xi_{a}-\eta_{a})/\sqrt{\epsilon}$ are the coordinates in the basis of simple roots. Taking this into account and passing to the limit $\epsilon\to 0$ in (5) we obtain the Gaussian distribution (6).

4.2. Plancherel distribution

Here we outline the proof of Theorem 3 under the assumption that the asymptotic of multiplicities is uniform in $\xi$ in $D_{0}$ including boundary strata555For a rigorous proof see [4] and [16]. In other words the asymptotic along the boundary can be obtained by taking corresponding limit in $\xi$.

For the Plancherel probability distribution we have the following asymptotic when $\epsilon\to 0$ and $\xi=\epsilon\lambda$ is finite and regular:

[TABLE]

We used the Weyl formula for the dimension of $V_{\lambda}$:

[TABLE]

The function $S(\tau,\xi)$ attains its maximum at $\xi=0$. So, we shall find the asymptotic of the formula (23) near this point. The convexity of the function $f$ implies that for small $\xi$, its Legendre dual $x$ is also small, therefore for small $\xi$

[TABLE]

Now let us compute the matrix of second derivative

[TABLE]

Here $\{H_{a}\}$ is a basis of simple roots in the Cartan subalgebra.

Lemma 1**.**

The following equality holds:

[TABLE]

where $B_{ab}$ is the symmetrized Cartan matrix and $c_{2}$ is the Casimir element (see below).

Proof.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

To find the numerator choose a basis $e_{i}$ in $\mathfrak{g}$. If $(\;.\;,\;.\;)$ is the Killing form and

[TABLE]

then for the Casimir element we have

[TABLE]

By Schur’s lemma

[TABLE]

for some $\tilde{x}_{\nu}$. This constant is easy to find taking the trace:

[TABLE]

which gives

[TABLE]

Assume that ${H_{a}}$ is part of the basis $e_{i}$, then

[TABLE]

Finally,

[TABLE]

∎

Denote $x=\frac{1}{dim(\mathfrak{g})}\sum_{k=1}^{m}\tau_{k}c_{2}(\nu_{k})$. As a consequence of the lemma above and the asymptotical formula for $x_{a}$ in terms of $\xi_{a}$ we have

[TABLE]

Thus, for small $\xi$ we have

[TABLE]

Recall that the matrix $K$ is defined as

[TABLE]

Lemma 1 implies that

[TABLE]

Thus, $K=B/x$ and for small $\xi$ we have

[TABLE]

Rescaling random variable $\xi$ as $\xi_{b}=\sqrt{\epsilon x}a_{b}$ we obtain the following asymptotic for the probabilities in Plancherel distribution:

[TABLE]

Let $h$ be a continuous bounded function on $\mathbb{R}^{r}_{\geq 0}$. The asymptotic (27) implies the convergence

[TABLE]

where the density function $p(a)$ is given in (9). In other words, the Plancherel measure weakly converges to $p(a)da_{1}\dots da_{r}$.

5. The asymptotic for $\mathfrak{g}=sl_{n+1}$, $m=1$, $\nu=\omega_{1}$ via the hook length formula.

In this section we recall the derivation of the pointwise asymptotic of multiplicities and of the Plancherel probability measure using the Schur-Weyl duality and the hook length formula from [9].

5.1. The asymptotic of multiplicity

Let us first review known results on multiplicities of irreducible $sl_{n+1}$ modules in the $N$-th tensor power of the vector representation. Due to the Schur-Weyl duality we have

[TABLE]

where $V_{\lambda}^{(n+1)}$ is the irreducible $sl_{n+1}$ module with the highest weight $\lambda$ corresponding to the partition $l_{1}\geq\dots\geq\l_{n+1}\geq 0$, $\sum_{i=1}^{n+1}l_{i}=N$ and in the simple root basis $\lambda_{a}=l_{a}-l_{a+1}$. The space $W_{\lambda}^{(N)}$ is an irreducible $S_{N}$ module corresponding to the partition $l$; $m_{\lambda}^{(N)}=dim(W_{\lambda}^{(N)})$ is the multiplicity of $V_{\lambda}\hookrightarrow(\mathbb{C}^{n+1})^{\otimes N}$. Here the Lie algebra $sl_{n+1}$ acts diagonally and the symmetric group $S_{N}$ permutes the factors.

The multiplicity function $m_{\lambda}^{(N)}$ (or the dimension of $W_{\lambda}^{(N)}$) is determined by the hook length formula, see, for example [10]:

[TABLE]

Using the Stirling formula

[TABLE]

we obtain the following asymptotics for multiplicities for large $N$ and $l_{i}$:

[TABLE]

Now, assume $N=\tau/\epsilon,\;\;l_{i}=\sigma_{i}/\epsilon$, where $\tau,\sigma_{i}$ are finite and $\epsilon\to 0$. Note that $\tau,\sigma_{i}$ satisfy the condition $\sum_{i=1}^{n+1}\sigma_{i}=\tau,\;\;\;\tau\geq\sigma_{1}\geq\dots\geq\sigma_{n+1}\geq 0$.

Then in the limit $\epsilon\to 0$ we obtain the asymptotic of the multiplicity function:

[TABLE]

where

[TABLE]

In the section 6 we will show that this formula matches the one from Theorem 1.

5.2. The asymptotic of the Plancherel probability distribution.

Now let us find the asymptotical probability distribution for the Plancherel probability measure which was first studied by Kerov in [9].

[TABLE]

5.2.1.

Dimensions of irreducible $sl_{n+1}$-modules are given by the Weyl formula

[TABLE]

First, let us find the asymptotic of $p_{\lambda}$ when $N=\tau/\epsilon,\;\;l_{i}=\sigma_{i}/\epsilon$, $\epsilon\to 0$ and $\tau,\sigma_{i}$ remain finite with $\sum_{i=1}^{n+1}\sigma_{i}=\tau,\;\;\;\tau\geq\sigma_{1}\geq\dots\geq\sigma_{n+1}\geq 0$. When $\epsilon\to 0$ we have

[TABLE]

Combining this with the asymptotic of the multiplicity we obtain the following pointwise asymptotic of $p_{\lambda}$:

[TABLE]

5.2.2.

Now, let us first find the maximum of the large deviation rate function $S(\tau,\sigma)$ on the hypersurface $\sum_{i=1}^{n+1}\sigma_{i}=\tau$. For this we can use the method of Lagrange multipliers and consider

[TABLE]

Critical points of $S(\tau,\sigma)$ are solutions to $\frac{\partial S_{\alpha}}{\partial\sigma_{i}}=0$ and $\frac{\partial S_{\alpha}}{\partial\alpha}=0$ which gives:

[TABLE]

This system has unique solution

[TABLE]

Now let us study the behaviour of $m_{\lambda}^{N}$ in the vicinity of this critical point. For this rescale random variables $\sigma_{i}$ as:

[TABLE]

Since $\sigma_{i}$ are constrained (30), we should have $\sum_{i=1}^{n+1}a_{i}=0$ and since $\sigma_{i}\geq\sigma_{i+1}\geq 0$ we should have $a_{i}\geq a_{i+1}$. Expanding $S(\tau,\sigma)$ in the Taylor series around $\sigma_{i}=\tau/(n+1)$ we have

[TABLE]

For the pointwise asymptotic of the asymptotic Plancherel probability distribution in the vicinity of the critical point we obtain

[TABLE]

This implies that the probability measure $p_{\lambda}$ converges weakly to the probability distribution of the hyperplane $\sum_{i}a_{i}=0$ in $\mathbb{R}^{n+1}$ with the density function

[TABLE]

This is exactly the result of [9].

6. The asymptotic of multiplicities for $\mathfrak{g}=sl_{n+1}$, $m=1,\nu=\omega_{1}$ via character identities.

Here we will show that for $N$-the tensor power of the vector representation of $sl_{n+1}$ the asymptotic of the multiplicity $m_{\lambda}^{N}$ derived from Theorem 1

[TABLE]

coincides with the asymptotic obtained from the hook length formula.

6.1. The large deviation rate function $S(\tau,\xi)$

For $sl_{n+1}$ the simple roots are $\alpha_{i}=e_{i}-e_{i+1},\;\;i=1\dots n$. The first fundamental weight is $\omega_{1}=\frac{1}{n+1}\left(ne_{1}-e_{2}-e_{3}-\dots-e_{n}\right)$. Weights of the first fundamental representation are given by

[TABLE]

where $\mu_{1}=\omega_{1}$ is the highest weight.

The character of the first fundamental module:

[TABLE]

where $(y,\mu_{i})$ is the scalar product in $\mathbb{R}^{n+1}$ and $y=y_{1}\alpha_{1}+y_{2}\alpha_{2}+\dots+y_{n}\alpha_{n}$.

According to the proposed method, the large deviation rate function for the first fundamental representation is given by the following expression:

[TABLE]

where $x$ is determined from the system of equations (19):

[TABLE]

Lemma 2**.**

The solution of system (34) is determined by

[TABLE]

where $\chi=\chi_{\omega_{1}}$ denotes the character of the first fundamental module.

Proof.

Firstly, we write the equations of (34) explicitly:

[TABLE]

To solve (35) let us introduce the variables $z_{i}=e^{x_{i}}$. So (35) can be written as:

[TABLE]

From the first equation we can express $z_{1}^{-1}z_{2}$ in terms of $z_{1}$ and then substitute it into the next equation. Step by step, we can express $z_{i}^{-1}z_{i+1}$ in terms of $z_{1}$:

[TABLE]

We then need to express $z_{1}$ in terms of $\chi$. This can be obtained by noting that

[TABLE]

Substiting $z_{i}^{-1}z_{i+1}$ from (36) into (37) we get

[TABLE]

Now, we can solve (36) in terms of $\chi$ and ${\xi_{i}}$:

[TABLE]

Finally, we can solve (35)in terms of $\chi$ and ${\xi_{i}}$:

[TABLE]

∎

Lemma 3**.**

The large deviation rate function (33) is given by

[TABLE]

Proof.

The large deviation rate function in terms of $\chi$ and ${\xi_{i}}$:

[TABLE]

To obtain expression for $\ln\chi$ in terms of ${\xi_{i}}$ we substitute solutions of (39) into (37):

[TABLE]

which yields

[TABLE]

and

[TABLE]

Substituting $\ln\chi$ into the large deviation rate function we finally obtain:

[TABLE]

where

[TABLE]

∎

The expression (40) coincides with (29) which we obtained from hook length formula.

6.2. The matrix $K$ and its determinant

Now let us compute the determinant of $K(x)$ when $x$ is the Legendre dual to $\xi$.

We have:

[TABLE]

From here we derive

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From here we derive

[TABLE]

6.3. The factor $\Delta(x)e^{-(\rho,x)}$

Finally, we need to compute $\Delta(x)$ and $e^{-(\rho,x)}$ . The Weyl vector $\rho$ is the sum of fundamental weights $\omega_{i}$:

[TABLE]

which gives

[TABLE]

For $x=\sum_{i=1}^{n}x_{i}\alpha_{i}$, where $\alpha_{i}=e_{i}-e_{i+1}$, we have

[TABLE]

Lemma 4**.**

[TABLE]

Indeed, we can use (39) and (3) to rewrite $e^{x_{i}}$ in terms of $\tau$ and ${\xi_{i}}$ and then in terms of $\sigma$ we obtain:

[TABLE]

This proves lemma 4.

Lemma 5**.**

[TABLE]

Proof.

The Weyl denominator $\Delta(x)$ is the sum

[TABLE]

where $(-1)^{l(w)}$ is the signature of the permutation $w\in S_{n+1}$. We have:

[TABLE]

which implies

[TABLE]

To evaluate $\Delta$ we can bring the factor

[TABLE]

out of summation. The remaining polynomial is the Vandermond determinant:

[TABLE]

∎

6.4. The comparison

Combining the results of the previous sections we conclude that

[TABLE]

where the function $S(\tau,\xi)$ on the left side is defined as the Legendre transform of $\tau\ln(\chi_{\omega_{1}}(e^{x}))$ in $x$ and on the right it is given by (29) and is derived from the hook formula.

7. Intermediate scaling

In this section we consider the behavior of the character probability measure

[TABLE]

near $t=0$ in the limit when $\epsilon\to 0$, $N_{k}=\frac{\tau_{k}}{\epsilon}$,$\lambda=\sqrt{\frac{x}{\epsilon}}b$, and $t=\sqrt{\frac{\epsilon}{x}}u$666There is an analogue of such asymptotic for every non regular value of $t$ (when $t\subset S\in\mathfrak{h}^{\star}\geq 0$ where $S$ is a stratum of the boundary of the cone $\mathfrak{h}^{\star}_{\geq 0}$). In this case transverse directions to $S$ should be scaled as $\sqrt{\epsilon}$. where $b\in\mathfrak{h}^{*}_{\geq 0}$, $u\in\mathfrak{h}_{\geq 0}$ are regular.

Theorem 5**.**

The pointwise asymptotic of $p_{\lambda}^{(N)}$ when $\epsilon\to 0$ is

[TABLE]

Proof.

The pointwise asymptotic of the multiplicity function is given by Theorem 1. As it was shown in section 4.2, the variables $x_{a}$ in the formula for the asymptotic for small $\xi_{a}$ are

[TABLE]

where $x=\frac{1}{dim(\mathfrak{g})}\sum_{k=1}^{m}\tau_{k}c_{2}(\nu_{k})$ is as in section 4.2. In terms of the variable $b$ we have

[TABLE]

As a consequence for the Weyl denominator we have

[TABLE]

It is also easy to find the asymptotic of the character:

[TABLE]

For small $\xi$ the large deviation rate function is

[TABLE]

In terms of variable $b$ $\xi_{a}=\sqrt{\epsilon x}b_{a}$ and we have:

[TABLE]

For the matrix $K_{ab}$ as in section 4.2 we have:

[TABLE]

and therefore

[TABLE]

Finally, taking into account that $t=\sqrt{\frac{\epsilon}{x}}u$ we have

[TABLE]

Here we used the identity

[TABLE]

where we assume that $V$ is irreducible, $c_{2}(V)$ is the value of the second Casimir on $V$ and $(x,y)$ is the scalar product with respect to the Killing form. The identity follows from the Schur lemma.

Now after the substitution of these asymptotical expressions into (43) we obtain the desired asymptotic:

[TABLE]

∎

Let us check now that the function

[TABLE]

is the density of a probability measure on $\mathfrak{h}^{\star}\geq 0$ ($b=\sum_{a}b_{a}\alpha_{a},b_{a}\geq 0$). It is clear that $p(b,u)\geq 0$, so we only have to check the normalization. We naturally expect this to be true since $\sum_{\lambda}p^{(N)}_{\lambda}(t)=1$. The following lemma checks it explicitly.

Lemma 6**.**

The function $p(b,u)$ is the density of a probability measure on $\mathfrak{h}^{\star}\geq 0$, i.e.

[TABLE]

where $db_{1}\dots db_{r}$ is the Euclidean measure on $\mathbb{R}^{r}$.

Proof.

Let us calculate the normalization constant $N$ explicitly:

[TABLE]

It is easy to compute the Gaussian integral

[TABLE]

We have

[TABLE]

where $D_{u}=\prod_{\alpha\in\Delta_{+}}\left(\frac{\partial}{\partial u},\alpha\right)$.

The normalization constant $N$ now can be written as

[TABLE]

where

[TABLE]

The following identity is easy to check:

[TABLE]

Thus

[TABLE]

∎

Note that (45) implies a weak convergence of the sequence of character measures $\{p^{(N)}_{\lambda}(t)\}$ to $p(b,u)db$.

8. Nonlinear PDE for the rate function $S(\tau,\xi)$

Let $V=\oplus_{\mu}V(\mu)$ be the weight decomposition of $V$ and $d_{\mu}=\dim(V_{\mu})$. For the weight $\lambda$, which is well inside the main Weyl chamber, we have the following decomposition of the tensor product

[TABLE]

were $wt(V)$ is the set of weights which occur in the representation $V$.

Let $m_{\lambda}^{(N)}$ be the multiplicities in

[TABLE]

the decomposition (46) gives the difference equation for multiplicities

[TABLE]

where we assume again that $\lambda$ is well inside the main Weyl chamber.

Now, passing to the limit $\epsilon\to 0$ in this difference equation with $N=\frac{\tau}{\epsilon}$, $\lambda=\frac{\xi}{\epsilon}$ we derive the following nonlinear PDE for the large deviation rate function $S$:

[TABLE]

Here we used the basis of simple roots in $\mathfrak{h}^{*}$: $\mu=\sum_{a}\mu_{a}\alpha_{a},\;\;\xi=\sum_{a}\xi_{a}\alpha_{a}$.

The following proposition is a direct verification of the equation (47) for $S(\tau,\xi)$ determined by (17) .

Proposition 1**.**

The function $S(\tau,\xi)$ defined in (3) satisfies the differential equation (47).

Proof.

By definition

[TABLE]

where $x$ is the unique solution to the equation:

[TABLE]

Differentiating this with respect to $\tau$ we have

[TABLE]

i.e.

[TABLE]

From here we obtain

[TABLE]

On the other hand

[TABLE]

i.e.

[TABLE]

Thus, both sides of (47) are equal to $\chi_{\nu}(e^{x})$ and we verified (47). ∎

9. Conclusion

This paper demonstrats that in large tensor products the statistic of irreducible components with respect to the character distribution almost does not depend on which representations being multiplied. But it depends significantly on whether the parameter $t$ in the character distribution is generic or special.

Note that one can associate natural Markov process with the decomposition of tensor powers $V^{\otimes N}$ of a finite dimensional $\mathfrak{g}$-module $V$. The transition probabilities in such a process are

[TABLE]

where $\chi(e^{t})$ is the character of the representation $V$ evaluated at $e^{t}$ and $b_{\lambda,\mu}$ are multiplicities of irreducible components in the tensor product

[TABLE]

Note that if $\lambda$ is sufficiently inside the positive Weyl chamber, $b_{\lambda,\lambda-\mu}=d_{\mu}$ where $d_{\mu}$ is the multiplicity of weight $\mu$ in $V$.

The character probability distribution (2) is a result of the Markov evolution of character measure of the trivial representation:

[TABLE]

i.e.

[TABLE]

This paper can be regarded as a study of this Markov process777Or of its slight generalization when we consider $\otimes_{k}V_{\nu_{k}}^{\otimes N_{k}}$ instead of $V^{\otimes N}$. . This process also can be regarded as a random walk on a lattice domain. For results in this direction see [3]; also [13] and references therein. From this point of view the results on the week convergence of character measures can be interpreted as follows:

• •

For regular $t$, as $N\to\infty$, the expectation value of $\lambda$ behave as $\mathbb{E}(\lambda)=N\nabla\ln(\chi(e^{t}))(1+o(1)$ where $\nabla$ is the gradient (with respect to the Killing form on the weight space). As $N\to\infty$ the distribution converges to the Gaussian distribution around the expectation value with the dispersion behaving as $\sqrt{N}$.

• •

For $t=0$ and $N\to\infty$ the expectation value of $\lambda$ vanishes and the asymptotical distribution is given by (9) rescaled such that the dispersion is proportional to $\sqrt{N}$.

Similar interpretation can be given for the intermediate scaling.

Below we will outline some of the future directions and problems in the asymptotic representation theory that are naturally related to this paper.

• •

We studied the asymptotic of dimensions of irreducible components and the corresponding probability distributions when $t$ is either regular or maximally irregular, i.e. $t=0$. When $t$ is not regular the asymptotical measure is given by a different formula involving the centralizer of the action of $W$ on $\xi$. These results will be presented in a separate publication.

• •

Truncated tensor products, also known as fusion products appear in representation theory of quantum groups at roots of unity and plays an important role in constructing modular categories. The latter are instrumental in conformal filed theory and in topological quantum filed theory. The study of statistics of irreducible components for truncated tensor products is an important problem by itself and an important step in understanding the semiclassical limit of corresponding topological quantum field theory. See [7] and references therein for existing results on fusion products and corresponding random walks.

Other interesting problems expanding the results announced here are related to multiplicities of irreducible components in tensor products of Lie superalgebras and of finite groups of Lie type, i.e. groups similar to $SL_{n}(F_{p})$. In these cases, as well as in the case of quantum groups at roots of unity, tensor products of irreducibles have both irreducible components and blocks of irreducibles. The study of statistics of blocks in large tensor products is another interesting problem that is largely open for investigation.