Modules of the 0-Hecke algebra and quasisymmetric Schur functions

Abstract

We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of Schur functions known as quasisymmetric Schur functions. Furthermore, we classify combinatorially which of these 0-Hecke modules are indecomposable.From here, we establish that the natural equivalence relation arising from our 0-Hecke action has equivalence classes that are isomorphic to subintervals of the weak Bruhat order on the symmetric group. Focussing on the equivalence classes containing a canonical tableau we discover a new basis for the Hopf algebra of quasisymmetric functions, and use the cardinality of these equivalence classes to establish new enumerative results on truncated shifted reverse tableau studied by Panova and Adin-King-Roichman.Generalizing our 0-Hecke action to one on skew standard reverse composition tableaux, we derive 0-Hecke modules whose quasisymmetric characteristics are the skew quasisymmetric Schur functions of Bessenrodt et al. This enables us to prove a restriction rule that reflects the coproduct formula for quasisymmetric Schur functions, which in turn yields a quasisymmetric branching rule analogous to the classical branching rule for Schur functions.2010 Mathematics Subject Classification. Primary 05E05, 20C08; Secondary 05A05, 05A19, 05E10, 06A07, 16T05, 20F55. Bruhat order.The authors were supported in part by the National Sciences and Engineering Research Council of Canada. 1 2 VASU V. TEWARI AND STEPHANIE J. VAN WILLIGENBURG 5. 0-Hecke modules from SRCTs and quasisymmetric Schur functions 15 6. Source and sink tableaux, and the weak Bruhat order 16 7. The classification of tableau-cyclic and indecomposable modules 23 8. The canonical basis and enumeration of truncated shifted reverse tableaux 26 8.1. Dimensions of certain S α,Eα and truncated shifted reverse tableaux 28 9. Restriction rules and skew quasisymmetric Schur functions 31 10. Further avenues 35 References 35

Indecomposable 0-Hecke modules for extended Schur functions

“…As is the case for the dual immaculate quasisymmetric functions, but not the case for the quasisymmetric Schur functions, the modules X α for the extended Schur functions are indecomposable. We devote the remainder of the paper to establishing this fact, following the approach of [BBS`15] and [TvW15]. Let T sup α be the standard extended tableau of shape α whose entries in the ith row are the first α i integers larger than α 1`¨¨¨`αi´1 .…”

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

“…More information about H n (0) and its representations can be found in [10,34], and contemporary results can be found in [6,23,24,25,26,29]. Our interest in the 0-Hecke algebra stems from the authors' previous work in the context of providing a representation-theoretic interpretation for quasisymmetric Schur functions [45].…”

“…The proof of the above theorem in the context of SCTs involves a lengthy verification and relies on some preliminary lemmas [45]. The techniques we need vary slightly and we discuss the differences next.…”

“…Composition tableaux were introduced in [19] to define the basis of quasisymmetric Schur functions for the Hopf algebra of quasisymmetric functions. These functions are analogues of the ubiquitous Schur functions [41], have been studied in substantial detail recently [8,19,20,30,31,45], and have consequently been the genesis of an active new branch of algebraic combinatorics discovering Schur-like bases in quasisymmetric functions [1,5], type B quasisymmetric Schur functions [27,37], quasi-key polynomials [2,42] and quasisymmetric Grothendieck polynomials [35]. Just as Young tableaux play a crucial role in the combinatorics of Schur functions [40,43], composition tableaux are key to understanding the combinatorics of quasisymmetric Schur functions.…”

Hecke algebras with independent parameters

“…Duchamp, Hivert, and Thibon [10] constructed the quiver of the 0-Hecke algebra H n (0) of type A n−1 and showed that H n (0) is of infinite representation type for n ≥ 4. Tewari and van Willigenburg [31,32] and König [21] studied connections between H n (0) and a new basis of quasisymmetric functions called the quasisymmetric Schur functions. After further investigation of combinatorial aspects of the representation theory of H n (0) [15,16] using the correspondence of Krob and Thibon, we recently extended this correspondence from type A to type B and type D [18,19].…”