AN INVITATION TO THE GENERALIZED SATURATION CONJECTURE

NEW

Problem: Reconstructed statement. Let $\alpha,\beta,\mu$ be partitions of the same size. For $|\alpha|=|\beta|$, define the internal product by

$$s_\alpha*s_\beta=\sum_\gamma g_{\alpha\beta\gamma}s_\gamma,$$

where $g_{\alpha\beta\gamma}$ are Kronecker coefficients. Define $L_{\alpha,\beta}^{\mu}(q)$ by

$$s_\alpha*s_\beta=\sum_\mu L_{\alpha,\beta}^{\mu}(q)P_\mu(x;q),$$

with $P_\mu(x;q)$ the Hall--Littlewood functions. Equivalently,

$$L_{\alpha,\beta}^{\mu}(q)=\sum_\gamma g_{\alpha\beta\gamma}K_{\gamma\mu}(q).$$

For $N\ge0$, let $N\alpha=(N\alpha_1,N\alpha_2,\ldots)$. The conjecture is:

$$\sum_{N\ge0}L_{N\alpha,N\beta}^{N\mu}(q)t^N\in \mathbb Q(q,t).$$

This is exactly Conjecture 2.25 as quoted; the definition of $L$ is from the surrounding section.

Result: The conjecture is true.

If $|\alpha|=0$, then all partitions are empty and the series is $(1-t)^{-1}$. Assume henceforth $|\alpha|>0$. Put

$$a=\ell(\alpha),\qquad b=\ell(\beta),\qquad r=\ell(\mu),$$

and choose vector spaces $A,B,C$ of dimensions $a,b,r$. Let

$$R=\operatorname{Sym}(A\otimes B\otimes C),$$

with the natural action of $G=\mathrm{GL}(A)\times\mathrm{GL}(B)\times\mathrm{GL}(C)$, and let $U\subset G$ be the product of the standard maximal unipotent subgroups.

By Schur--Weyl duality,

$$R \cong \bigoplus_{\lambda,\nu,\gamma} (\mathbb S_\lambda A\otimes \mathbb S_\nu B\otimes \mathbb S_\gamma C)^{\oplus g_{\lambda\nu\gamma}}.$$

Hence the multigraded Hilbert series

$$H(X,Y,Z)=\sum_{\lambda,\nu,\gamma}g_{\lambda\nu\gamma}X^\lambda Y^\nu Z^\gamma$$

is the character of $R^U$. By Grosshans finite generation, $R^U$ is finitely generated, so Hilbert--Serre gives $H$ as a rational function with denominator a finite product of binomials $1-X^\lambda Y^\nu Z^\gamma$.

Now use Kostant’s formula for Kostka--Foulkes polynomials. Let

$$\Pi_q(U)=\prod_{1\le i<j\le r}(1-qU_i/U_j)^{-1} =\sum_\eta p_q(\eta)U^\eta.$$

For partitions $\gamma,N\mu$ of the same size and length at most $r$,

$$K_{\gamma,N\mu}(q) =\sum_{w\in S_r}(-1)^{\ell(w)} p_q\bigl(w(\gamma+\delta)-N\mu-\delta\bigr),$$

where $\delta=(r-1,r-2,\ldots,0)$. No terms are lost by restricting to $\ell(\gamma)\le r$, since $K_{\gamma,N\mu}(q)\ne0$ implies $\gamma\trianglerighteq N\mu$, hence $\ell(\gamma)\le r$.

For each $w\in S_r$, define

$$\Phi_w= U^{\delta-w\delta}\Pi_q(U)\, H\bigl(X,Y,Z_j=U_{w(j)}^{-1}\bigr)\, \frac1{1-tX^{-\alpha}Y^{-\beta}U^\mu}.$$

Taking the constant term in $X,Y,U$, expansion gives

$$\operatorname{CT}\Phi_w = \sum_{N\ge0}\sum_\gamma g_{N\alpha,N\beta,\gamma}\, p_q\bigl(w(\gamma+\delta)-N\mu-\delta\bigr)t^N.$$

Therefore

$$\sum_{N\ge0}L_{N\alpha,N\beta}^{N\mu}(q)t^N = \sum_{w\in S_r}(-1)^{\ell(w)}\operatorname{CT}\Phi_w.$$

It remains only to note that each $\operatorname{CT}\Phi_w$ is rational in $q,t$. Indeed, after writing $H$ by Hilbert--Serre, every $\Phi_w$ is a rational function whose denominator is a finite product of terms $1-M$, with $M$ a Laurent monomial in the constant-term variables times a monomial in $q,t$. Expanding these factors, the constant term counts integer solutions of a finite system of linear equations in nonnegative variables, weighted by monomials in $q,t$. By Gordan’s lemma, such affine semigroup generating functions are rational. Thus each $\operatorname{CT}\Phi_w\in\mathbb Q(q,t)$, and the finite alternating sum is rational.

Verification audit. The proof uses exactly the $L_{\alpha,\beta}^{\mu}(q)$ defined by Hall--Littlewood expansion, includes $N=0$, imposes no nonzero or saturation hypothesis, and applies Kostant’s formula only after verifying the required length bound. The conclusion is precisely rationality of $\sum_{N\ge0}L_{N\alpha,N\beta}^{N\mu}(q)t^N$.

Citation: No prior source is invoked as proving Conjecture 2.25 exactly. Standard ingredients used: Macdonald, Symmetric Functions and Hall Polynomials, Ch. I §7 and Ch. III §6; Grosshans, Algebraic Homogeneous Spaces and Invariant Theory, LNM 1673; Stanley, Enumerative Combinatorics, Vol. 1, on affine semigroup rational generating functions.