Equivalence classes of lower and upper descent weak Bruhat intervals

초록

Let Int(n) denote the set of nonempty left weak Bruhat intervals in the symmetric group S-n. We investigate the equivalence relation on Int(n), where I similar or equal to(D)J if and only if there exists a descent-preserving poset isomorphism between I and J. For each equivalence class C of (Int(n), similar or equal to(D)), a partial order <= is defined by [sigma,rho](L) <= [sigma ',rho '](L) if and only if sigma <= R sigma '. Kim-Lee-Oh (2024) showed that the poset (C,<=) is isomorphic to a right weak Bruhat interval. In this paper, we focus on lower and upper descent weak Bruhat intervals, specifically those of the form [omega(0)(S),sigma](L) or [sigma,omega(1)(S)](L), where omega(0)(S) is the longest element in the parabolic subgroup SS of Sn, generated by {si|(i is an element of S)} for a subset S subset of [n-1], and omega(1)(S) is the longest element among the minimal-length representatives of left S[n-1]\S-cosets in S-n. We begin by providing a poset-theoretic characterization of the equivalence relation. Using this characterization, the minimal and maximal elements within an equivalence class C are identified when C is a lower or upper descent interval. Under an additional condition, a detailed description of the structure of (C,<=) is provided. Furthermore, for the equivalence class containing [omega(0)(S),sigma]L, an injective hull of B([omega(0)(S),sigma]L) is given, and for the equivalence class containing [sigma,omega(1)(S)]L, a projective cover of B([sigma,omega(1)(S)]L) is given. Here, B(I) denotes the weak Bruhat interval module of the 0-Hecke algebra associated with I is an element of Int(n). The results obtained are applied to investigate lower descent intervals arising from quotient modules and submodules of projective indecomposable modules of the 0-Hecke algebra. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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