Projective Resolutions of Generic Order Ideals

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

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Promotion and Evacuation

The Electronic Journal of Combinatorics ◽

10.37236/75 ◽

2009 ◽

Vol 16 (2) ◽

Cited By ~ 17

Author(s):

Richard P. Stanley

Keyword(s):

Vector Space ◽

Hecke Algebra ◽

Linear Extensions ◽

Linear Transformations ◽

Basic Properties ◽

Finite Poset ◽

Order Ideals ◽

Maximal Chains

Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schützenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset $P$ may be regarded as maximal chains in the lattice $J(P)$ of order ideals of $P$. The generalizations concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of ${\Bbb F}_q^n$, then the results can be stated in terms of the expansion of certain Hecke algebra products.

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Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets

The Electronic Journal of Combinatorics ◽

10.37236/4334 ◽

2016 ◽

Vol 23 (1) ◽

Cited By ~ 11

Author(s):

Darij Grinberg ◽

Tom Roby

Keyword(s):

Finite Order ◽

Disjoint Union ◽

Parallel Analysis ◽

Birational Map ◽

Finite Poset ◽

Order Ideals ◽

Set Up ◽

Grafting Onto

We study a birational map associated to any finite poset $P$. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of $P$. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call "skeletal". Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and "grafting onto an antichain"; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion.

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Regular projective resolutions OF SEMIMODULES

HUE UNIVERSITY JOURNAL OF SCIENCE NATURAL SCIENCE ◽

10.26459/jns.v126i1b.4272 ◽

2017 ◽

Vol 126 (1B) ◽

Author(s):

Nguyễn Xuân Tuyến

Keyword(s):

Comparison Theorem ◽

Projective Resolution ◽

Projective Resolutions

In this paper, we present an approach version of semimodule homologies by regular projective resolutions such as define a concept of a regular projective resolution, prove the comparison theorem for semimodules by these resolutions and based them provide cohomology monoids of semimodules.

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