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Bases of the quantum matrix bialgebra and induced sign characters of the Hecke algebra

Authors
  • Kaliszewski, Ryan1
  • Lambright, Justin2
  • Skandera, Mark1
  • 1 Lehigh University, Department of Mathematics, Bethlehem, PA, USA , Bethlehem (United States)
  • 2 Anderson University, School of Science and Engineering, Anderson, IN, USA , Anderson (United States)
Type
Published Article
Journal
Journal of Algebraic Combinatorics
Publisher
Springer US
Publication Date
Jun 28, 2018
Volume
49
Issue
4
Pages
475–505
Identifiers
DOI: 10.1007/s10801-018-0832-4
Source
Springer Nature
License
Yellow

Abstract

We combinatorially describe entries of the transition matrices which relate monomial bases of the zero-weight space of the quantum matrix bialgebra. This description leads to a combinatorial rule for evaluating induced sign characters of the type A Hecke algebra Hn(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_n(q)$$\end{document} at all elements of the form (1+Tsi1)⋯(1+Tsim)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1 + T_{s_{i_1}}) \cdots (1 + T_{s_{i_m}})$$\end{document}, including the Kazhdan–Lusztig basis elements indexed by 321-hexagon-avoiding permutations. This result is the first subtraction-free rule for evaluating all elements of a basis of the Hn(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_n(q)$$\end{document}-trace space at all elements of a basis of Hn(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_n(q)$$\end{document}.

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