Two stability theorems on plethysms of Schur functions

TL;DR

This work tackles the long-standing problem of decomposing plethysm products of Schur functions by proving two stability theorems for plethysm coefficients under adding or joining partitions to either the outer partition $ u$ or the inner skew partition $oldsymbol{ extmu}/oldsymbol{ extmu}_igstar$. The proofs are purely combinatorial, built around plethystic semistandard signed tableaux and a Signed Weight Lemma that reduces stability to the existence of stable partition systems in a twisted dominance framework. The authors also extend known stability results to skew shapes, provide explicit bounds for stability and zero-stability, and develop substantial theory around twisted Kostka numbers and insertion bijections (the $ ext{ extfont{F}}$ map) to realize stable correspondences. Their approach unifies prior results (including Foulkes-type and Law–Okitani–Brion-type stability) and offers a robust, computationally amenable framework, with software tools to explore the twisted-interval structures. The results have broad implications for understanding polynomial GL$_n$-representations and the combinatorics of plethysm in a stable regime, with potential applications to positivity questions and representation theory of wreath products and symmetric groups.

Abstract

The plethysm product of Schur functions corresponds to composing polynomial representations of infinite general linear groups. Finding the plethysm coefficients $\langle s_ν\circ s_μ, s_λ\rangle$ that express an arbitrary plethysm $s_ν\circ s_μ$ as a sum $\sum_λ\langle s_ν\circ s_μ, s_λ\rangle s_λ$ of Schur functions is a fundamental open problem in algebraic combinatorics. We prove two stability theorems for plethysm coefficients under the operations of adding and/or joining an arbitrary partition to either $μ$ or $ν$. In both theorems $μ$ may be replaced with an arbitrary skew partition. As special cases we obtain all stability results on the plethysm product of two Schur functions in the literature to date. The proofs are entirely combinatorial using plethystic semistandard tableaux with positive and negative entries.

Figures (12)

• Figure 3.1: The skew partition $(6,5,5,2)/(3,1)$ above is $(4,3)$-large in the sense of Definition \ref{['defn:large']} because $(3,4) \in [(6,5,5,2)]$. It is also $(5,3)$-large, but not $(5,4)$-large.
• Figure 6.1: The partitions in the $\ell^{-}$-decomposition $\langle\sigma^{{-}}, \sigma^{+}\rangle$ of $\sigma \in \mathrm{Par}$. Note that $\sigma^{-}$ has at most $\ell^{-}$ parts and that $\sigma^{-}_{\ell^{-}} \ge \ell(\sigma^{+})$, so $\sigma$ is $\bigl( \ell(\sigma^{-})+1, \ell(\sigma^{+}) \bigr)$-large in the sense of Definition \ref{['defn:large']}, having $\bigl( \ell(\sigma^{+}), \ell(\sigma^{-}) + 1 \bigr)$ as the shaded box. In particular $\sigma$ is $\bigl( \ell(\sigma^{-}), \ell(\sigma^{+}) \bigr)$-large.
• Figure 6.2: Hasse diagram of the up-set $(6,2)^{\,\unlhd\space\raisebox{1pt}{$\cdot$}\space}$ in the $1$-twisted dominance order on $\mathrm{Par}(8)$, as seen in \ref{['eq:62upset']} in the overview of the proof in §\ref{['sec:overview']}. By Remark \ref{['remark:twistedDominanceOrderGeneralizesDominanceOrder']}, this up-set is also the interval $[(6,2),(1^8)]_{\hbox{$\unlhd\space\raisebox{1pt}{$\cdot$}$}\,}$. The total order ${\,\le\space\raisebox{1pt}{$\cdot$}\,}$ refining ${\,\unlhd\space\raisebox{1pt}{$\cdot$}\space}$ is indicated by vertical height. The matrix with entries $|\mathop{\mathrm{SSYT}}\nolimits(\sigma)_{(\pi^{-},\pi^{+})}|$ for $\pi$, $\sigma \in (6,2)^{\hbox{$\unlhd\space\raisebox{1pt}{$\cdot$}$}\,}$ relevant to condition (b) in the definition of a stable partition system (Definition \ref{['defn:stablePartitionSystem']}) is shown to the right, with row and column labels ordered by the total order in Definition \ref{['defn:twistedTotalOrder']}. It is lower unitriangular by Lemma \ref{['lemma:twistedKostkaMatrix']}. We use $\cdot$ to show a zero implied by this lemma.
• Figure 7.1: Hasse diagrams of up-sets in the $2$-twisted dominance order. The total order ${\,\le\space\raisebox{1pt}{$\cdot$}\,}$ refining ${\,\unlhd\space\raisebox{1pt}{$\cdot$}\space}$ defined in Definition \ref{['defn:twistedTotalOrder']} is indicated by vertical height. On the left is the up-set of $(4,4,4) \!\leftrightarrow\! \bigl\langle(3,3), (2,2,2)\bigr\rangle$ restricted to partitions of length at most $3$. (This is part of the up-set relevant to Example \ref{['ex:444support']} and the following remark.) This poset maps under $\lambda \mapsto \lambda \space\oplus\space \bigl( (1,1), (2) \bigr)$ into the up-set of $(6,4,4,2) \!\leftrightarrow\! \bigl\langle(4,4), (4,2,2)\bigr\rangle$ restricted to partitions of length at most $4$, shown in the middle; the two partitions not in the image of the map are highlighted. In turn, for each $M \ge 1$, the middle poset is in bijection, by iterating this map, with the up-set of $(4,4,4) \oplus M\bigl((1^2),(2)) = (4+2M,4,4,2^M) \!\leftrightarrow\! \bigl\langle(3+M,3+M), (2+2M,2,2)\bigr\rangle$ cut to partitions of length at most $M+3$, as shown on the right. (To save space we write $M'$ for $M+3$.)
• Figure 7.2: The two semistandard signed tableaux in the sets $\mathop{\mathrm{SSYT}}\nolimits\bigl( (7,5,2,2) \bigr)_{((4,4),(4,2,2))}$ and $\mathop{\mathrm{SSYT}}\nolimits\bigl( (9,5,2,2,2) \bigr)_{((5,5),(6,2,2))}$. The hatched boxes are inserted by the $\mathcal{F}$ insertion map.

Theorems & Definitions (229)

• Theorem 1.1: Signed inner stability
• Theorem 1.2: Signed outer stability
• Conjecture 1.3
• Remark 2.1
• Definition 3.1
• Remark 3.2
• Definition 3.3: Signed tableau
• Definition 3.4: Signed weight
• Definition 3.5: Signed weight of a signed tableau
• Definition 3.6: Sign of a signed tableau