scholarly journals The rook monoid is lexicographically shellable

2019 ◽  
Vol 81 ◽  
pp. 265-275
Author(s):  
Mahir Bilen Can
Keyword(s):  
Rook Monoid

REPRESENTATIONS OF THE SYMPLECTIC ROOK MONOID

2008 ◽  
Vol 18 (05) ◽  
pp. 837-852 ◽  
Author(s):  
ZHENHENG LI ◽  
ZHUO LI ◽  
YOU'AN CAO
Keyword(s):  
Weyl Group ◽  
Symmetric Groups ◽  
Triangular Matrix ◽  
Character Table ◽  
Irreducible Representations ◽  
Rook Monoid ◽  
Algebraic Description ◽  
Practical Algorithm ◽  
Upper Triangular Matrix ◽  
User Friendly

In this paper, we concern representations of symplectic rook monoids R. First, an algebraic description of R as a submonoid of a rook monoid is obtained. Second, we determine irreducible representations of R in terms of the irreducible representations of certain symmetric groups and those of the symplectic Weyl group W. We then give the character formula of R using the character of W and that of the symmetric groups. A practical algorithm is provided to make the formula user-friendly. At last we show that the Munn character table of R is a block upper triangular matrix.

THE ANNIHILATOR OF TENSOR SPACE IN THE -ROOK MONOID ALGEBRA

2017 ◽  
Vol 96 (1) ◽  
pp. 77-86
Author(s):  
ZHANKUI XIAO
Keyword(s):  
Explicit Construction ◽  
Tensor Space ◽  
Rook Monoid

In this paper, we give an explicit construction of a quasi-idempotent in the $q$-rook monoid algebra $R_{n}(q)$ and show that it generates the whole annihilator of the tensor space $U^{\otimes n}$ in $R_{n}(q)$.

THE BRAID ROOK MONOID

2008 ◽  
Vol 18 (04) ◽  
pp. 779-802 ◽  
Author(s):  
EDDY GODELLE
Keyword(s):  
Group Theory ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Hecke Algebra ◽  
Linear Algebraic Group ◽  
Weyl Groups ◽  
Rook Monoid ◽  
Algebraic Monoid ◽  
Tits Group

In linear algebraic monoid theory, the Renner monoids play the role of the Weyl groups in linear algebraic group theory. It is well known that Weyl groups are Coxeter groups, and that we can associate a Hecke algebra and an Artin–Tits group to each Coxeter group. The question of the existence of a Hecke algebra associated with each Renner monoid has been positively answered. In this paper we discuss the question of the existence of an equivalent of the Artin–Tits groups in the framework of Renner monoids. We consider the seminal case of the rook monoid and introduce a new length function.

scholarly journals PRESENTATION FOR RENNER MONOIDS

2010 ◽  
Vol 83 (1) ◽  
pp. 30-45 ◽  
Author(s):  
EDDY GODELLE
Keyword(s):  
Length Function ◽  
Monoid Presentation ◽  
Rook Monoid ◽  
Renner Monoid

AbstractWe extend the result obtained in E. Godelle [‘The braid rook monoid’, Internat. J. Algebra Comput.18 (2008), 779–802] to every Renner monoid: we provide a monoid presentation for Renner monoids, and we introduce a length function which extends the Coxeter length function and which behaves nicely.

scholarly journals Jucys–Murphy elements of partition algebras for the rook monoid

2021 ◽  
pp. 1-34
Author(s):  
Ashish Mishra ◽  
Shraddha Srivastava
Keyword(s):  
Representation Theory ◽  
Irreducible Representations ◽  
Spectral Approach ◽  
Inductive Approach ◽  
Partition Algebra ◽  
Rook Monoid ◽  
Partition Algebras ◽  
Weyl Duality ◽  
Multiplicity Free

Kudryavtseva and Mazorchuk exhibited Schur–Weyl duality between the rook monoid algebra [Formula: see text] and the subalgebra [Formula: see text] of the partition algebra [Formula: see text] acting on [Formula: see text]. In this paper, we consider a subalgebra [Formula: see text] of [Formula: see text] such that there is Schur–Weyl duality between the actions of [Formula: see text] and [Formula: see text] on [Formula: see text]. This paper studies the representation theory of partition algebras [Formula: see text] and [Formula: see text] for rook monoids inductively by considering the multiplicity free tower [Formula: see text] Furthermore, this inductive approach is established as a spectral approach by describing the Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin bases, determined by the aforementioned multiplicity free tower, of irreducible representations of [Formula: see text] and [Formula: see text]. Also, we describe the Jucys–Murphy elements of [Formula: see text] which play a central role in the demonstration of the actions of Jucys–Murphy elements of [Formula: see text] and [Formula: see text].

scholarly journals Characters of Renner monoids and their Hecke algebras

2020 ◽  
Vol 30 (07) ◽  
pp. 1505-1535
Author(s):  
Andrew Hardt ◽  
Jared Marx-Kuo ◽  
Vaughan McDonald ◽  
John M. O’Brien ◽  
Alexander Vetter
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra ◽  
Explicit Description ◽  
General Algorithm ◽  
Character Table ◽  
Cartan Type ◽  
Type Formula ◽  
Rook Monoid ◽  
Combinatorial Formulas ◽  
Renner Monoid

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e. the Cartan type [Formula: see text] Renner monoid) was computed earlier by Dieng et al. [2], using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type [Formula: see text] and use them to obtain an explicit description of the character table for the type [Formula: see text] Renner monoid Hecke algebra.

scholarly journals Gelfand Models for Diagram Algebras

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Tom Halverson
Keyword(s):  
Symmetric Group ◽  
Linear Span ◽  
Linear Representation ◽  
Irreducible Representations ◽  
Semisimple Algebra ◽  
Diagram Algebras ◽  
Rook Monoid ◽  
International Audience ◽  
Complex Linear ◽  
Diagram Algebra

International audience A Gelfand model for a semisimple algebra $\mathsf{A}$ over $\mathbb{C}$ is a complex linear representation that contains each irreducible representation of $\mathsf{A}$ with multiplicity exactly one. We give a method of constructing these models that works uniformly for a large class of combinatorial diagram algebras including: the partition, Brauer, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, and planar rook monoid algebras. In each case, the model representation is given by diagrams acting via ``signed conjugation" on the linear span of their vertically symmetric diagrams. This representation is a generalization of the Saxl model for the symmetric group, and, in fact, our method is to use the Jones basic construction to lift the Saxl model from the symmetric group to each diagram algebra. In the case of the planar diagram algebras, our construction exactly produces the irreducible representations of the algebra.

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