[Paper Review] Shifted tableau switchings and shifted Littlewood-Richardson coefficients

Seung-Il Choi, Sun-Young Nam+1 more|arXiv (Cornell University)|Jan 31, 2017

Advanced Combinatorial Mathematics10 references3 citations

TL;DR

This paper introduces two new combinatorial algorithms—shifted tableau switching and modified shifted tableau switching—as analogues of the classical tableau switching process for shifted Young tableaux. By defining specialized elementary transformations (switches) that preserve shifted perforateness and lattice word properties, the authors establish an involutive, bijective map that provides a combinatorial interpretation of shifted Littlewood-Richardson coefficients, proving symmetry and giving a constructive rule for Schur $P$- and $Q$-function identities.

ABSTRACT

We provide two shifted analogues of the tableau switching process due to Benkart, Sottile, and Stroomer, the shifted tableau switching process and the modified shifted tableau switching process. They are performed by applying a sequence of specially contrived elementary transformations called {\em switches} and turn out to have some spectacular properties. For instance, the maps induced from these algorithms are involutive and behave very nicely with respect to shifted Young tableaux whose reading words satisfy the lattice property. As an application, we give combinatorial interpretations of Schur $P$- and $Q$-function identities. We also demonstrate the relationship between the shifted tableau switching process and the shifted $J$-operation due to Worley.

Motivation & Objective

• To extend the classical tableau switching process to the shifted setting, where tableaux are defined on shifted shapes and include primed entries.
• To resolve the issue that standard shifted jeu de taquin can produce non-semistandard tableaux by introducing modified switches that preserve semistandardness.
• To provide a combinatorial interpretation of Schur $P$- and $Q$-function identities using the new switching algorithms.
• To establish a bijection between pairs of shifted tableaux satisfying the lattice property, thereby giving a constructive proof of the symmetry of shifted Littlewood-Richardson coefficients.
• To demonstrate that the modified shifted tableau switching is an involution and preserves the lattice property, enabling a new rule for computing shifted Littlewood-Richardson coefficients.

Proposed method

• Define shifted perforated $({f a},{f b})$-pairs as a framework for handling the switching process on shifted tableaux with empty boxes.
• Introduce seven elementary switches (S1–S7) that mimic the action of shifted jeu de taquin and allow the exchange of entries in a controlled way.
• Construct the shifted tableau switching process by sequentially applying switches to transform a pair $(S,T)$ of shifted tableaux with $T$ extending $S$ into a new pair $({}^S T, S_T)$, maintaining shifted perforateness.
• Introduce three modified switches (S1′, S2′, S6′) to prevent primed entries from appearing on the main diagonal, ensuring semistandardness is preserved.
• Define the modified shifted tableau switching process using the modified switches in specific cases and standard switches otherwise, ensuring the output remains a pair of semistandard shifted tableaux.
• Prove that the modified switching map is an involution and preserves the lattice property of reading words, using shifted Knuth equivalence and prime manipulation invariance.

Experimental results

Research questions

• RQ1 Can the classical tableau switching process be generalized to the shifted setting, preserving key properties like involutivity and rectification behavior?
• RQ2 How can the issue of non-semistandard outputs in shifted jeu de taquin be resolved to allow for a well-defined combinatorial rule for shifted Littlewood-Richardson coefficients?
• RQ3 Is there a bijective algorithm that provides a combinatorial interpretation of the symmetry $g^{ u}_{ ueta} = g^{ u}_{etaeta}$ for shifted Littlewood-Richardson coefficients?
• RQ4 What conditions ensure that the reading word of a shifted tableau remains a lattice word under switching operations?
• RQ5 Can the modified shifted tableau switching process be used to derive a constructive rule for the expansion of Schur $P$-functions in terms of Schur $P$-functions?

Key findings

• The modified shifted tableau switching process is an involution, meaning applying it twice returns the original pair of tableaux.
• The process preserves the lattice property of reading words: a tableau has a lattice word if and only if its image under switching does.
• The number of modified Littlewood-Richardson-Stembridge (LRS) tableaux of shape $ u/ u$ and weight $ u$ is exactly $g^{ u}_{ ueta} = rac{2^{ u} f^{ u}_{etaeta}}{2^{ u - eta}}$, with an explicit example showing $g^{ u}_{ ueta} = 8$ for $ u = (6,5,2,1)$, $eta = (4,2)$, $ u = (4,3,1)$.
• The modified shifted tableau switching induces a bijection between $igcup_{ u i eta} ext{SSYT}( u/eta, u) imes ext{SSYT}(eta)$ and $igcup_{ u i u} ext{SSYT}( u) imes ext{SSYT}( u/eta, u)$, providing a combinatorial proof of the identity $P_{ u/eta}(x) = igsum_{ u} g^{ u}_{ ueta} P_{ u}(x)$.
• The symmetry of the shifted Littlewood-Richardson coefficients is not only proven but explicitly realized via an involution that swaps the inner shape and weight of a modified LRS tableau, with $g^{ u}_{ ueta} = g^{ u}_{eta u}$.