[Paper Review] Reduced words for clans

Brian Burks, Brendan Pawlowski|arXiv (Cornell University)|Jun 13, 2018

Advanced Combinatorial Mathematics20 references3 citations

TL;DR

This paper establishes a Matsumoto-Tits-type theorem for reduced words in the weak order on clans—combinatorial objects indexing orbits of $\mathrm{GL}(\mathbb{C}^p)\times\mathrm{GL}(\mathbb{C}^q)$ on flag varieties. It introduces an equivalence relation on words that captures reduced word sets for clans, proves that these sets are unions of equivalence classes, and provides enumerative formulas linking reduced words to standard and shifted standard tableaux, with exact counts via symmetric functions and hook-length formulas.

ABSTRACT

Clans are combinatorial objects indexing the orbits of $GL(\mathbb{C}^p) imes GL(\mathbb{C}^q)$ on the variety of flags in $\mathbb{C}^{p+q}$. This geometry leads to a partial order on the set of clans analogous to weak Bruhat order on the symmetric group, and we study the saturated chains in this order. We prove an analogue of the Matsumoto-Tits theorem on reduced words in a Coxeter group. We also obtain enumerations of reduced word sets for particular clans in terms of standard tableaux and shifted standard tableaux.

Motivation & Objective

• To develop a combinatorial framework for reduced words in the weak order on clans, generalizing the Matsumoto-Tits theorem from Coxeter groups.
• To characterize the structure of reduced word sets for clans using an equivalence relation on words that respects the orbit geometry.
• To enumerate reduced words for specific clans using standard and shifted standard tableaux.
• To establish symmetric function identities involving Schur functions and Schur Q-functions that encode the number of maximal chains in the clan poset.

Proposed method

• Define a partial order on clans analogous to weak Bruhat order, with covering relations labeled by adjacent transpositions in $[n-1]$.
• Introduce an equivalence relation $\equiv$ on words over $[n-1]$ generated by Coxeter relations and a sign-flip rule: $a_1\cdots a_\ell \equiv (n-a_1)a_2\cdots a_\ell$, capturing symmetries of the clan geometry.
• Use symmetric functions $F_\gamma$ associated to clans $\gamma$, where the coefficient of a monomial counts reduced words for $\gamma$, extending Stanley's framework for permutations.
• Establish a bijection between matchless clans and partitions inside $[p]\times[q]$, linking $F_\gamma$ to products of Schur functions $s_{\lambda^+}s_{\lambda^-}$.
• Apply the shifted hook-length formula to compute the number of unmarked standard shifted tableaux of shape $\lambda = (n-1,n-3,\ldots,n-2q+1)$, which counts maximal chains.
• Prove that $\sum_{\gamma \text{ matchless}} F_\gamma = 2^q P_{(n-1,n-3,\ldots,n-2q+1)} = Q_{(n-1,n-3,\ldots,n-2q+1)}$, linking the sum to Schur Q-functions.

Experimental results

Research questions

• RQ1 Can a Matsumoto-Tits-type theorem be extended from Coxeter groups to the weak order on clans, despite reduced word sets not being single equivalence classes?
• RQ2 How do the reduced word sets for clans relate to symmetric functions, and can they be enumerated via standard and shifted standard tableaux?
• RQ3 What is the exact number of maximal chains in the weak order on $(p,q)$-clans, and how is it encoded in symmetric function identities?
• RQ4 What is the relationship between the combinatorics of matchless clans and shifted Young tableaux, and how does this yield a symmetric function identity?
• RQ5 Under what conditions does the symmetric function $2^{\kappa(z)}\hat{F}_z$ equal a single Schur Q-function, and how does this relate to the enumeration of reduced words?

Key findings

• The equivalence relation $\equiv$ on words, generated by Coxeter relations and the sign-flip rule $a_1\cdots a_\ell \equiv (n-a_1)a_2\cdots a_\ell$, ensures that each reduced word set $\mathcal{R}(\gamma)$ is a union of equivalence classes, and this is the strongest such relation.
• For any clan $\gamma$, the set of reduced words $\mathcal{R}(\gamma)$ is a union of equivalence classes under $\equiv$, and $a \equiv b$ if and only if they are in the same reduced word sets for all clans.
• The sum of symmetric functions $\sum_{\gamma \text{ matchless}} F_\gamma$ equals $2^q P_{(n-1,n-3,\ldots,n-2q+1)}$, which is also equal to the Schur Q-function $Q_{(n-1,n-3,\ldots,n-2q+1)}$, establishing a symmetric function identity.
• The number of maximal chains in $\mathrm{Clan}_{p,q}$ is $2^{pq} g^{\lambda}$, where $g^{\lambda}$ is the number of unmarked standard shifted tableaux of shape $\lambda = (n-1,n-3,\ldots,n-2q+1)$, computed via the shifted hook-length formula.
• The coefficient of $x_1x_2\cdots x_{pq}$ in $\sum_{\gamma \text{ matchless}} F_\gamma$ is $2^{pq} g^{\lambda}$, which matches the number of maximal chains, confirming the enumerative formula.
• The map $\gamma \mapsto \lambda^+ (\gamma)$ gives a bijection between matchless $(p,q)$-clans and partitions inside $[p]\times[q]$, and $\lambda^-(\gamma)^t = \lambda^+(\gamma)^\vee$, leading to the identity $\sum_{\lambda \subseteq [p]\times[q]} s_\lambda s_{\lambda^{\vee t}} = Q_{(p+q-1,p+q-3,\ldots,p-q+1)}$.