[Paper Review] Generating series of cyclically fully commutative elements is rational

Mathias Pétréolle|arXiv (Cornell University)|Dec 12, 2016

semigroups and automata theory2 references3 citations

TL;DR

This paper proves that the generating series for cyclically fully commutative (CFC) elements in any Coxeter group is rational by constructing a finite state automaton that recognizes reduced expressions of CFC elements. Using this automaton and a result on minimal reduced decompositions, the authors establish rationality through closure properties of regular languages.

ABSTRACT

In this paper, we study the generating function of cyclically fully commutative elements in Coxeter groups, which are elements such that any cyclic shift of theirs reduced decompositions remains a reduced expression of a fully commutative element. By designing a finite state automaton recognizing reduced expressions of cyclically fully commutative elements, we can show that the aforementioned generating series is always rational.

Motivation & Objective

• To establish the rationality of the generating series for cyclically fully commutative (CFC) elements in arbitrary Coxeter groups.
• To extend prior results on rationality of generating series for fully commutative elements to the broader class of CFC elements.
• To provide a constructive method for computing the generating series using automata-theoretic techniques.
• To generalize previous results on finite and affine Coxeter groups to arbitrary Coxeter systems.

Proposed method

• Design a finite state automaton that recognizes the language of reduced expressions of CFC elements in a Coxeter group.
• Prove the correctness of the automaton by verifying that it accepts exactly the reduced expressions of CFC elements using combinatorial properties of reduced words and cyclic shifts.
• Leverage Brink and Howlett's result on minimal reduced decompositions being recognizable by finite automata.
• Intersect the automaton for minimal reduced decompositions with the automaton for CFC expressions to isolate exactly one representative per CFC element.
• Use closure properties of regular languages under intersection and the rationality of generating series for regular languages to conclude rationality of the generating series.
• Implement the automaton algorithmically to compute regular expressions and generate the series, enabling computational verification.

Experimental results

Research questions

• RQ1 Is the generating series for cyclically fully commutative elements rational in all Coxeter groups?
• RQ2 Can a finite state automaton be constructed to recognize reduced expressions of CFC elements?
• RQ3 Does the intersection of the minimal reduced decomposition language with the CFC expression language yield a regular language?
• RQ4 Can the rationality of the generating series be established without a direct recursive decomposition of the CFC elements?
• RQ5 What is the structure and size of the automaton recognizing CFC expressions, and is it minimal?

Key findings

• The generating series $ W^{CFC}(x) $ for cyclically fully commutative elements in any Coxeter group is rational.
• A finite state automaton recognizing reduced expressions of CFC elements is explicitly constructed and proven correct.
• The intersection of the language of minimal reduced decompositions and the language of CFC expressions yields a regular language, which implies rationality of the generating series.
• The method provides an algorithmic procedure to compute regular expressions for CFC elements and their generating series.
• The result confirms and generalizes earlier findings on finite and affine Coxeter groups, where the generating series were already known to be rational.
• The automaton for CFC expressions has a large number of states (e.g., 149 for $\widetilde{A}_3$), suggesting potential for optimization or minimization.