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

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

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

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

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

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

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,

Hence the multigraded Hilbert series

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

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

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

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

Therefore

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.