Rationality conjecture for the branching generating function of finite subgroups of $\mathbf{SL}_n\mathbb{C}$

About 23 years old · traced to 2003

Let $n\geq 2$ and let $\Gamma$ be a subgroup of $\mathbf{SL}_n\mathbb{C}$ . Let $\{\gamma_0,\ldots,\gamma_l\}$ be the equivalence classes of irreducible finite-dimensional complex representations of $\Gamma$ , with $\gamma_0$ trivial. For the irreducible $\mathfrak{sl}_n\mathbb{C}$ -module of highest weight $p_1\varpi_1+\cdots+p_{n-1}\varpi_{n-1}$ , write its restriction to $\Gamma$ as

\left.\pi_{\mathbf p}\right|_{\Gamma}=\bigoplus_{i=0}^l m_i(\mathbf p)\gamma_i,

where $\mathbf p=(p_1,\ldots,p_{n-1})\in\mathbb N^{n-1}$ . Set $\mathbf t^{\mathbf p}=t_1^{p_1}\cdots t_{n-1}^{p_{n-1}}$ , let $\mathcal E=(e_0,\ldots,e_l)$ be the canonical basis of $\mathbb C^{l+1}$ , and define

v_{\mathbf p}:=\sum_{i=0}^l m_i(\mathbf p)e_i,\qquad P_\Gamma(\mathbf t):=\sum_{\mathbf p\in\mathbb N^{n-1}}v_{\mathbf p}\mathbf t^{\mathbf p}.

Write $P_\Gamma(\mathbf t)_i$ for the $i$ -th coordinate. Rationality conjecture. The coefficients of the vector $P_\Gamma(\mathbf t)$ are rational functions in $\mathbf t$ : for every $i\in\{0,\ldots,l\}$ , there are polynomials $N_\Gamma(\mathbf t)_i,D_\Gamma(\mathbf t)\in\mathbb Q[\mathbf t]$ such that

P_\Gamma(\mathbf t)_i=\frac{N_\Gamma(\mathbf t)_i}{D_\Gamma(\mathbf t)}.

This conjecture generalizes the known results of Kostant for $\mathbf{SL}_2\mathbb C$ and the authors' results for $\mathbf{SL}_3\mathbb C$ ; it asserts rationality of the multivariable branching generating functions for the restrictions of irreducible representations to $\Gamma$ .

References

Primary source

Frédéric Butin, “Branching Law for the Finite Subgroups of SL(4,C)”, arXiv:1307.2557 (2013).

Additional references

2 papers in this index state this conjecture (2003–2013). The statement above is taken from the most recent of them; the others are arXiv:math/0309440.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Open

A reader-submitted argument claims to prove the conjecture for finite groups, but no independent verification or published resolution was found.

The conjecture asserts rationality of all multivariable branching generating functions for restrictions of irreducible representations of finite subgroups of $\mathbf{SL}_n(\mathbb{C})$ . The catalogued literature records the cases of $\mathbf{SL}_2(\mathbb{C})$ and $\mathbf{SL}_3(\mathbb{C})$ , but no general resolution.

Known results

Kostant established the $\mathbf{SL}_2(\mathbb{C})$ case.
The authors of the later catalogued paper established the $\mathbf{SL}_3(\mathbb{C})$ case.

Community submission (unverified), September 5, 2026

A submitted proof argues that, for finite $\Gamma$ with exponent $e$ , every branching series has a common denominator $\prod_{j=1}^{n-1}(1-t_j^e)^{\binom{n}{j}}$ . It uses Schur-polynomial character generation, alternation, and evaluation on finite-group eigenvalues; the argument is unverified.

Current status (as of September 2026): The $\mathbf{SL}_2(\mathbb{C})$ and $\mathbf{SL}_3(\mathbb{C})$ cases are recorded, while the general conjecture remains open on the retrieved evidence and the September 5, 2026 submission is unverified.

Sources

ProofAn AI-assisted derivation of Butin’s rationality conjecture from classical character-generating-function theory, with an explicit common integer denominator.See full solutionHide full solution

Conjecture 7 follows from the classical character-generating-function method of Patera and Sharp, reviewed by N. Okeke and M. A. Walton, On character generators for simple Lie algebras (2007), §2, equations (11)–(12) and (22). Here is a self-contained specialization to finite subgroups, including the case of repeated eigenvalues.

Let $\Gamma\subset SL_n(\mathbb C)$ be finite and let $e$ be its exponent. Write $\chi_i$ for an irreducible character of $\Gamma$ . We prove that the denominator

Q(t)=\prod_{j=1}^{n-1}(1-t_j^e)^{\binom nj}

works simultaneously for all the branching series $P_i(t)$ , with integer polynomial numerators.

For independent variables $z_1,\ldots,z_n$ , put

D(z,t)=\prod_{j=1}^{n-1}\prod_{\substack{S\subseteq\{1,\ldots,n\}\\|S|=j}} \left(1-t_j\prod_{a\in S}z_a\right).

For $p\in\mathbb N_0^{n-1}$ , set $\lambda_b(p)=\sum_{j=b}^{n-1}p_j$ for $b<n$ , and $\lambda_n(p)=0$ . The character of the $SL_n$ -representation of highest weight $\sum_jp_j\omega_j$ is the Schur polynomial $s_{\lambda(p)}(z)$ on determinant-one diagonal matrices.

Let $\Delta(z)=\det(z_a^{n-b})_{a,b=1}^n=\prod_{a<b}(z_a-z_b)$ . Expanding the bialternant formula and summing the geometric series in the $p_j$ gives

F(z,t):=\sum_p s_{\lambda(p)}(z)t^p =\frac1{\Delta(z)}\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma) \frac{\prod_{b=1}^n z_{\sigma(b)}^{n-b}} {\prod_{j=1}^{n-1}\left(1-t_j\prod_{b=1}^jz_{\sigma(b)}\right)}.

Every denominator in this finite sum divides $D$ , so multiplying its alternating numerator sum by $D$ gives a polynomial $B\in\mathbb Z[z,t]$ . Since $D$ is symmetric in $z$ , $B$ is alternating. Consequently every $z_a-z_b$ divides $B$ ; these are distinct prime factors in $\mathbb Z[z,t]$ , so their product $\Delta$ divides $B$ . Hence $A:=B/\Delta$ is an integer polynomial and

F(z,t)=\frac{A(z,t)}{D(z,t)}.

Because $D(z,0)=1$ , this is an identity in $\mathbb Z[z][[t]]$ . It can therefore be evaluated at any complex $z$ , including coincident eigenvalues.

For $g\in\Gamma$ , let $z_1,\ldots,z_n$ be its eigenvalues. The matrix $g$ is diagonalizable and $z_a^e=1$ for every $a$ . For each $j$ , every factor of $D(z,t)$ is $1-\zeta t_j$ for an $e$ -th root of unity $\zeta$ , and there are $\binom nj$ factors counting multiplicity. Since $1-t_j^e=\prod_{\zeta^e=1}(1-\zeta t_j)$ , the specialized $D(z,t)$ divides $Q(t)$ in $\mathbb C[t]$ . Therefore $Q(t)F_g(t)$ is a polynomial, where

F_g(t)=\sum_p\chi_{\lambda(p)}(g)t^p.

Finite-group character orthogonality now yields

P_i(t)=\frac1{|\Gamma|}\sum_{g\in\Gamma}\overline{\chi_i(g)}F_g(t).

It follows that $Q(t)P_i(t)\in\mathbb C[t]$ . But $P_i(t)$ has integer coefficients and $Q(t)\in\mathbb Z[t]$ , so this polynomial actually lies in $\mathbb Z[t]$ . Thus $P_i=N_i/Q$ with $N_i,Q\in\mathbb Z[t]$ , proving Butin's Conjecture 7, §3.3. $\square$

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