[Paper Review] Schensted-type correspondences and plactic monoids for types $B_{n}$ and $D_{n}$

Cédric Lecouvey|ArXiv.org|Nov 28, 2002

Algebraic structures and combinatorial models5 references4 citations

TL;DR

This paper establishes Schensted-type correspondences and defines plactic monoids for types $B_n$ and $D_n$ using Kashiwara's crystal basis theory. It introduces insertion algorithms for orthogonal and spin tableaux, derives defining relations for the monoids $\mathrm{Pl}(B_n)$ and $\mathrm{Pl}(D_n)$, and constructs a Jeu de Taquin for type $B$ based on Sheats' algorithm, proving that two words are equivalent in the monoid if and only if they lie in the same connected component of the crystal graph.

ABSTRACT

We use Kashiwara's theory of crystal bases to study plactic monoids for $U_{q}(so_{2n+1})$ and $U_{q}(so_{2n})$. Simultaneously we describe a Schensted type correspondence in the crystal graphs of tensor powers of vector and spin representations and we derive a Jeu de Taquin for type $B$ from the Sheats sliding algorithm.

Motivation & Objective

• To extend the Schensted correspondence and plactic monoid construction from type $A_n$ to types $B_n$ and $D_n$ using crystal basis theory.
• To define and characterize the plactic monoids $\mathrm{Pl}(B_n)$ and $\mathrm{Pl}(D_n)$ through equivalence relations on words in the alphabets $\mathcal{B}_n^*$ and $\mathcal{D}_n^*$.
• To develop insertion algorithms for orthogonal and spin tableaux that yield Schensted-type correspondences in the crystal graphs of tensor powers of vector and spin representations.
• To generalize the Jeu de Taquin for type $B$ using Sheats' sliding algorithm for type $C$, and to extend the framework to include spin representations.

Proposed method

• Use Kashiwara's crystal basis theory to analyze the crystal graphs of tensor powers of the fundamental representations of $U_q(so_{2n+1})$ and $U_q(so_{2n})$.
• Introduce spin columns to model the spin representations $V(\Lambda_n^B)$, $V(\Lambda_n^D)$, and $V(\Lambda_{n-1}^D)$, and define extended alphabets $\mathfrak{B}_n = \mathcal{B}_n \cup SP_n$ and $\mathfrak{D}_n = \mathcal{D}_n \cup SP_n$.
• Define equivalence relations $\overset{B}{\sim}$ and $\overset{D}{\sim}$ on words based on coherency in isomorphic connected components of the extended crystal graphs $\mathfrak{G}_n^B$ and $\mathfrak{G}_n^D$, and prove that $\mathrm{Pl}(B_n) = \mathcal{B}_n^* / \overset{B}{\sim}$ and $\mathrm{Pl}(D_n) = \mathcal{D}_n^* / \overset{D}{\sim}$ are monoids.
• Construct column insertion algorithms for orthogonal tableaux (analogous to Young tableaux for type $B$ and $D$) using the Kashiwara-Nakashima framework.
• Define oscillating tableaux (analogous to standard tableaux) to realize the $Q$-symbol in the Schensted correspondence, with shapes changing by one box at each step.
• Extend the insertion and equivalence framework to include spin representations by defining generalized tableaux $\mathfrak{P}(w)$ and $\mathfrak{Q}(w)$ for words in $\mathfrak{B}_n^*$ and $\mathfrak{D}_n^*$, and prove that $w_1 \sim w_2$ iff $w_1 \equiv w_2$ in the monoid.

Experimental results

Research questions

• RQ1 How can the Schensted correspondence be generalized to types $B_n$ and $D_n$ using crystal basis theory and insertion algorithms?
• RQ2 What are the defining relations for the plactic monoids $\mathrm{Pl}(B_n)$ and $\mathrm{Pl}(D_n)$, and how do they relate to the equivalence of words in the crystal graphs?
• RQ3 Can a Jeu de Taquin for type $B$ be constructed using Sheats' sliding algorithm for type $C$, and how does it relate to the insertion process?
• RQ4 How do spin representations affect the structure of the crystal graphs and the equivalence relations in the plactic monoids?
• RQ5 What is the relationship between the $P$-symbol and $Q$-symbol in the extended Schensted correspondence for types $B$ and $D$?

Key findings

• The plactic monoid $\mathrm{Pl}(B_n)$ is isomorphic to $\mathcal{B}_n^* / \overset{B}{\sim}$, where $w_1 \overset{B}{\sim} w_2$ if and only if $w_1$ and $w_2$ lie in the same isomorphic connected component of the crystal graph $\mathfrak{G}_n^B$, and similarly for $\mathrm{Pl}(D_n)$.
• The insertion algorithm for orthogonal tableaux yields a Schensted-type correspondence in $G_n^B$ and $G_n^D$, mapping each word $w$ to a pair $(\mathfrak{P}(w), \mathfrak{Q}(w))$ with $\mathfrak{P}(w)$ a generalized orthogonal tableau and $\mathfrak{Q}(w)$ an oscillating tableau.
• For any word $w$ in $\mathfrak{B}_n^*$ or $\mathfrak{D}_n^*$, the generalized tableau $\mathfrak{P}(w)$ is uniquely determined by the insertion process, and $w \sim w'$ if and only if $\mathfrak{Q}(w) = \mathfrak{Q}(w')$, meaning they lie in the same connected component.
• The extended monoids $\mathfrak{Pl}(B_n)$ and $\mathfrak{Pl}(D_n)$ are defined as $\mathfrak{B}_n^* / \overset{B}{\sim}$ and $\mathfrak{D}_n^* / \overset{D}{\sim}$, and the equivalence $w_1 \sim w_2$ holds if and only if $w_1 \equiv w_2$ in the monoid.
• A Jeu de Taquin for type $B$ is constructed based on Sheats' algorithm for type $C$, and it preserves the structure of the insertion process and the $Q$-symbol correspondence.
• For highest weight vertices $w_1$ and $w_2$ of $\mathfrak{G}_n$, $w_1 \sim w_2$ if and only if $w_1 \equiv w_2$, and $\mathfrak{P}(w)$ is the unique generalized orthogonal tableau such that $\mathrm{w}(\mathfrak{P}(w)) \sim w$.