1,661 research outputs found
Combinatorial Markov chains on linear extensions
We consider generalizations of Schuetzenberger's promotion operator on the
set L of linear extensions of a finite poset of size n. This gives rise to a
strongly connected graph on L. By assigning weights to the edges of the graph
in two different ways, we study two Markov chains, both of which are
irreducible. The stationary state of one gives rise to the uniform
distribution, whereas the weights of the stationary state of the other has a
nice product formula. This generalizes results by Hendricks on the Tsetlin
library, which corresponds to the case when the poset is the anti-chain and
hence L=S_n is the full symmetric group. We also provide explicit eigenvalues
of the transition matrix in general when the poset is a rooted forest. This is
shown by proving that the associated monoid is R-trivial and then using
Steinberg's extension of Brown's theory for Markov chains on left regular bands
to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in
terms of discrete time Markov chain
Flag-symmetry of the poset of shuffles and a local action of the symmetric group
We show that the poset of shuffles introduced by Greene in 1988 is
flag-symmetric, and we describe a "local" permutation action of the symmetric
group on the maximal chains which is closely related to the flag symmetric
function of the poset. A key tool is provided by a new labeling of the maximal
chains of a poset of shuffles, which is also used to give bijective proofs of
enumerative properties originally obtained by Greene. In addition we define a
monoid of multiplicative functions on all posets of shuffles and describe this
monoid in terms of a new operation on power series in two variables.Comment: 34 pages, 6 figure
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
Application of graph combinatorics to rational identities of type A
To a word , we associate the rational function . The main object, introduced by C. Greene to generalize
identities linked to Murnaghan-Nakayama rule, is a sum of its images by certain
permutations of the variables. The sets of permutations that we consider are
the linear extensions of oriented graphs. We explain how to compute this
rational function, using the combinatorics of the graph . We also establish
a link between an algebraic property of the rational function (the
factorization of the numerator) and a combinatorial property of the graph (the
existence of a disconnecting chain).Comment: This is the complete version of the submitted fpsac paper (2009
Gr\"obner methods for representations of combinatorial categories
Given a category C of a combinatorial nature, we study the following
fundamental question: how does the combinatorial behavior of C affect the
algebraic behavior of representations of C? We prove two general results. The
first gives a combinatorial criterion for representations of C to admit a
theory of Gr\"obner bases. From this, we obtain a criterion for noetherianity
of representations. The second gives a combinatorial criterion for a general
"rationality" result for Hilbert series of representations of C. This criterion
connects to the theory of formal languages, and makes essential use of results
on the generating functions of languages, such as the transfer-matrix method
and the Chomsky-Sch\"utzenberger theorem.
Our work is motivated by recent work in the literature on representations of
various specific categories. Our general criteria recover many of the results
on these categories that had been proved by ad hoc means, and often yield
cleaner proofs and stronger statements. For example: we give a new, more
robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb),
and a family of natural generalizations, are noetherian; we give an easy proof
of a generalization of the Lannes-Schwartz artinian conjecture from the study
of generic representation theory of finite fields; we significantly improve the
theory of -modules, introduced by Snowden in connection to syzygies of
Segre embeddings; and we establish fundamental properties of twisted
commutative algebras in positive characteristic.Comment: 41 pages; v2: Moved old Sections 3.4, 10, 11, 13.2 and connected text
to arxiv:1410.6054v1, Section 13.1 removed and will appear elsewhere; v3:
substantial revision and reorganization of section
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