907 research outputs found
The biHecke monoid of a finite Coxeter group
The usual combinatorial model for the 0-Hecke algebra of the symmetric group
is to consider the algebra (or monoid) generated by the bubble sort operators.
This construction generalizes to any finite Coxeter group W. The authors
previously introduced the Hecke group algebra, constructed as the algebra
generated simultaneously by the bubble sort and antisort operators, and
described its representation theory.
In this paper, we consider instead the monoid generated by these operators.
We prove that it has |W| simple and projective modules. In order to construct a
combinatorial model for the simple modules, we introduce for each w in W a
combinatorial module whose support is the interval [1,w] in right weak order.
This module yields an algebra, whose representation theory generalizes that of
the Hecke group algebra. This involves the introduction of a w-analogue of the
combinatorics of descents of W and a generalization to finite Coxeter groups of
blocks of permutation matrices.Comment: 12 pages, 1 figure, submitted to FPSAC'1
Spectral gap for random-to-random shuffling on linear extensions
In this paper, we propose a new Markov chain which generalizes
random-to-random shuffling on permutations to random-to-random shuffling on
linear extensions of a finite poset of size . We conjecture that the second
largest eigenvalue of the transition matrix is bounded above by
with equality when the poset is disconnected. This Markov
chain provides a way to sample the linear extensions of the poset with a
relaxation time bounded above by and a mixing time of . We conjecture that the mixing time is in fact as for the
usual random-to-random shuffling.Comment: 16 pages, 10 figures; v2: typos fixed plus extra information in
figures; v3: added explicit conjecture 2.2 + Section 3.6 on the diameter of
the Markov Chain as evidence + misc minor improvements; v4: fixed
bibliograph
On the representation theory of finite J-trivial monoids
In 1979, Norton showed that the representation theory of the 0-Hecke algebra
admits a rich combinatorial description. Her constructions rely heavily on some
triangularity property of the product, but do not use explicitly that the
0-Hecke algebra is a monoid algebra.
The thesis of this paper is that considering the general setting of monoids
admitting such a triangularity, namely J-trivial monoids, sheds further light
on the topic. This is a step to use representation theory to automatically
extract combinatorial structures from (monoid) algebras, often in the form of
posets and lattices, both from a theoretical and computational point of view,
and with an implementation in Sage.
Motivated by ongoing work on related monoids associated to Coxeter systems,
and building on well-known results in the semi-group community (such as the
description of the simple modules or the radical), we describe how most of the
data associated to the representation theory (Cartan matrix, quiver) of the
algebra of any J-trivial monoid M can be expressed combinatorially by counting
appropriate elements in M itself. As a consequence, this data does not depend
on the ground field and can be calculated in O(n^2), if not O(nm), where n=|M|
and m is the number of generators. Along the way, we construct a triangular
decomposition of the identity into orthogonal idempotents, using the usual
M\"obius inversion formula in the semi-simple quotient (a lattice), followed by
an algorithmic lifting step.
Applying our results to the 0-Hecke algebra (in all finite types), we recover
previously known results and additionally provide an explicit labeling of the
edges of the quiver. We further explore special classes of J-trivial monoids,
and in particular monoids of order preserving regressive functions on a poset,
generalizing known results on the monoids of nondecreasing parking functions.Comment: 41 pages; 4 figures; added Section 3.7.4 in version 2; incorporated
comments by referee in version
Directed nonabelian sandpile models on trees
We define two general classes of nonabelian sandpile models on directed trees
(or arborescences) as models of nonequilibrium statistical phenomena. These
models have the property that sand grains can enter only through specified
reservoirs, unlike the well-known abelian sandpile model.
In the Trickle-down sandpile model, sand grains are allowed to move one at a
time. For this model, we show that the stationary distribution is of product
form. In the Landslide sandpile model, all the grains at a vertex topple at
once, and here we prove formulas for all eigenvalues, their multiplicities, and
the rate of convergence to stationarity. The proofs use wreath products and the
representation theory of monoids.Comment: 43 pages, 5 figures; introduction improve
Markov chains, -trivial monoids and representation theory
We develop a general theory of Markov chains realizable as random walks on
-trivial monoids. It provides explicit and simple formulas for the
eigenvalues of the transition matrix, for multiplicities of the eigenvalues via
M\"obius inversion along a lattice, a condition for diagonalizability of the
transition matrix and some techniques for bounding the mixing time. In
addition, we discuss several examples, such as Toom-Tsetlin models, an exchange
walk for finite Coxeter groups, as well as examples previously studied by the
authors, such as nonabelian sandpile models and the promotion Markov chain on
posets. Many of these examples can be viewed as random walks on quotients of
free tree monoids, a new class of monoids whose combinatorics we develop.Comment: Dedicated to Stuart Margolis on the occasion of his sixtieth
birthday; 71 pages; final version to appear in IJA
Learn to Move Through a Combination of Policy Gradient Algorithms: DDPG, D4PG, and TD3
Bach N, Melnik A, Schilling M, Korthals T, Ritter H. Learn to Move Through a Combination of Policy Gradient Algorithms: DDPG, D4PG, and TD3. In: 6th International Conference, LOD 2020, Siena, Italy, Proceedings. Lecture Notes in Computer Science. Springer; 2020
The relationship between cognitive ability and personality scores in selection situations: A meta‐analysis
Several faking theories have identified applicants’ cognitive ability (CA) as a determinant of faking—the intentional distortion of answers by candidates—but the corresponding empirical findings in the area of personality tests are often ambiguous. Following the assumption that CA is important for faking, we expected applicants with high CA to show higher personality scores in selection situations, leading in this case to significant correlations between CA and personality scores, but not in nonselection situations. This meta‐analysis (66 studies, k = 115 individual samples, N = 46,265) showed this pattern of results as well as moderation effects for the study design (laboratory vs. field), the response format of the personality test, and the type of CA test
The biHecke monoid of a finite Coxeter group and its representations
For any finite Coxeter group W, we introduce two new objects: its cutting
poset and its biHecke monoid. The cutting poset, constructed using a
generalization of the notion of blocks in permutation matrices, almost forms a
lattice on W. The construction of the biHecke monoid relies on the usual
combinatorial model for the 0-Hecke algebra H_0(W), that is, for the symmetric
group, the algebra (or monoid) generated by the elementary bubble sort
operators. The authors previously introduced the Hecke group algebra,
constructed as the algebra generated simultaneously by the bubble sort and
antisort operators, and described its representation theory. In this paper, we
consider instead the monoid generated by these operators. We prove that it
admits |W| simple and projective modules. In order to construct the simple
modules, we introduce for each w in W a combinatorial module T_w whose support
is the interval [1,w]_R in right weak order. This module yields an algebra,
whose representation theory generalizes that of the Hecke group algebra, with
the combinatorics of descents replaced by that of blocks and of the cutting
poset.Comment: v2: Added complete description of the rank 2 case (Section 7.3) and
improved proof of Proposition 7.5. v3: Final version (typo fixes, picture
improvements) 66 pages, 9 figures Algebra and Number Theory, 2013. arXiv
admin note: text overlap with arXiv:1108.4379 by other author
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