Rook Monoid Research Articles

Schur–Weyl dualities for the rook monoid: an approach via Schur algebras

The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur–Weyl duality between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra. We also explain how this relates to Solomon’s Schur–Weyl duality between the rook monoid and the general linear group and mention some advantages of our approach.

An extension to “A subsemigroup of the rook monoid”

A recent paper studied an inverse submonoid Mn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{n}$$\\end{document} of the rook monoid, by representing the nonzero elements of Mn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{n}$$\\end{document} via certain triplets belonging to Z3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^3$$\\end{document}. In this note, we allow the triplets to belong to R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. We thus study a new inverse monoid M¯n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{M}_{n}$$\\end{document}, which is a supermonoid of Mn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{n}$$\\end{document}. We point out similarities and find essential differences. We show that M¯n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{M}_{n}$$\\end{document} is a noncommutative, periodic, combinatorial, fundamental, completely semisimple, and strongly E∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${E}^{*}$$\\end{document}-unitary inverse monoid.

Alternating submodules for partition algebras, rook algebras, and rook-Brauer algebras

Letting n≥2k, the partition algebra CAk≥2(n) has two one-dimensional subrepresentations that correspond in a natural way to the alternating and trivial characters of the symmetric group Sk. In 2019, Benkart and Halverson introduced and proved evaluations in the two distinguished bases of CAk(n) for nonzero elements in the one-dimensional regular CAk(n)-submodule that corresponds to the Young symmetrizer ∑σ∈Skσ; in 2016, Xiao proved an explicit formula for the analogue of the sign representation for the rook monoid algebra. In this article, we lift Xiao's formula to a diagram basis evaluation in the partition algebra CAk(n). We prove that our diagram basis evaluation for this lifting, which we denote as Altk∈CAk(n), generates a one-dimensional module under the action of multiplication by arbitrary elements in CAk(n). Our explicit formula for Altk gives us a cancellation-free formula for the other one-dimensional regular CAk(n)-module, with regard to Benkart and Halverson's lifting of ∑σ∈Skσ. We then use a sign-reversing involution to evaluate our one-dimensional generators in the orbit basis, and we use our explicit formula for Altk to lift Young's N- and P-functions so as to allow set-partition tableaux as arguments, and we use this lifting to construct Young-type matrix units for CA2(n) and CA3(n).

Semiring identities of finite inverse semigroups

We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial rook monoid admits no finite identity basis, and so do almost all additively idempotent semirings whose multiplicative reducts are combinatorial inverse semigroups.

A subsemigroup of the rook monoid

We define a subsemigroup \(S_n\) of the rook monoid \(R_n\) and investigate its properties. To do this, we represent the nonzero elements of \(S_n\) (which are \(n\times n\) matrices) via certain triplets of integers, and develop a closed-form expression representing the product of two elements; these tools facilitate straightforward deductions of a great number of properties. For example, we show that \(S_n\) consists solely of idempotents and nilpotents, find the numbers of idempotents and nilpotents, compute nilpotency indexes, determine Green’s relations and ideals, and come up with a minimal generating set. Furthermore, we give a necessary and sufficient condition for the jth root of a nonzero element to exist in \(S_n\), show that existence implies uniqueness, and compute the said root explicitly. We also point to several combinatorial aspects; describe a number of subsemigroups of \(S_n\) (some of which are familiar from previous studies); and, using rook n-diagrams, graphically interpret many of our results.

Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

We consider a tower of generalized rook monoid algebras over the field \mathbb{C} of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys–Murphy elements for generalized rook monoid algebras. Over an algebraically closed field \Bbbk of positive characteristic p , utilizing Jucys–Murphy elements of rook monoid algebras, for 0\leq i\leq p-1 we define the corresponding i -restriction and i -induction functors along with two extra functors. On the direct sum \mathcal{G}_{\mathbb{C}} of the Grothendieck groups of module categories over rook monoid algebras over \Bbbk , these functors induce an action of the tensor product of the universal enveloping algebra U(\widehat{\mathfrak{sl}}_p(\mathbb{C})) and the monoid algebra \mathbb{C}[\mathcal{B}] of the bicyclic monoid \mathcal{B} . Furthermore, we prove that \mathcal{G}_{\mathbb{C}} is isomorphic to the tensor product of the basic representation of U(\widehat{\mathfrak{sl}}_{p}(\mathbb{C})) and the unique infinite-dimensional simple module over \mathbb{C}[\mathcal{B}] , and also exhibit that \mathcal{G}_{\mathbb{C}} is a bialgebra. Under some natural restrictions on the characteristic of \Bbbk , we outline the corresponding result for generalized rook monoids.

Jucys–Murphy elements of partition algebras for the rook monoid

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].

Characters of Renner monoids and their Hecke algebras

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.

Sects

By explicitly describing a cellular decomposition we determine the Borel invariant cycles that generate the Chow groups of the quotient of a reductive group by a Levi subgroup. For illustration we consider the variety of polarizations SLn/S(GLp×GLq), and we introduce the notion of a sect for describing its cellular decomposition. In particular, for p=q, we show that the Bruhat order on the sect corresponding to the dense cell is isomorphic, as a poset, to the rook monoid with the Bruhat-Chevalley-Renner order.

Set-partition tableaux and representations of diagram algebras

The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis indexed by set-partition tableaux such that diagrams in the algebra act combinatorially on tableaux. The first representation is analogous to the Gelfand model and the second is a generalization of Young's natural representation of the symmetric group on standard tableaux. The methods of this paper work uniformly for the partition algebra and its diagram subalgebras. As an application, we express the characters of each of these algebras as nonnegative integer combinations of symmetric group characters whose coefficients count fixed points under conjugation.

The rook monoid is lexicographically shellable

We prove that the Bruhat–Chevalley–Renner order on the rook monoid is EL-shellable. We determine the homeomorphism type of the associated order complex.

THE ANNIHILATOR OF TENSOR SPACE IN THE -ROOK MONOID ALGEBRA

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 $R_{n}(q)$.

On tensor spaces for rook monoid algebras

Let $m,n\in \mathbb{N}$, and $V$ be a $m$-dimensional vector space over a field $F$ of characteristic $0$. Let $U=F\oplus V$ and $R_n$ be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of $U^{\otimes n}$ in $FR_n$, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of $U^{\otimes n}$ in $FR_n$.

Fourier Inversion for Finite Inverse Semigroups

This paper continues the study of Fourier transforms on finite inverse semigroups, with a focus on Fourier inversion theorems and FFTs for new classes of inverse semigroups. We begin by introducing four inverse semigroup generalizations of the Fourier inversion theorem for finite groups. Next, we describe a general approach to the construction of fast inverse Fourier transforms for finite inverse semigroups complementary to an approach to FFTs given in previous work. Finally, we give fast inverse Fourier transforms for the symmetric inverse monoid and its wreath product by arbitrary finite groups, as well as fast Fourier and inverse Fourier transforms for the planar rook monoid, the partial cyclic shift monoid, and the partial rotation monoid.

Enumeration of the Generalized Rook Monoid

The generalized rook monoid is introduced.The order of the generalized rook monoid is calculated and description of the generalized rook monoid is obtatined by matrices.

Gelfand models for diagram algebras

A Gelfand model for a semisimple algebra $$\mathsf {A}$$ A over an algebraically closed field $$\mathbb {K}$$ K is a linear representation that contains each irreducible representation of $$\mathsf {A}$$ A with multiplicity exactly one. We give a method of constructing these models that works uniformly for a large class of semisimple, 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 horizontally symmetric diagrams. This representation is a generalization of the Saxl model for the symmetric group. 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.

Pattern avoidance in the rook monoid

We consider two types of pattern avoidance in the rook monoid, i.e. the set of 0–1 square matrices with at most one nonzero entry in each row and each column. For one-dimensional rook patterns, we completely characterize monoid elements avoiding a single pattern of length at most three and develop an enumeration scheme algorithm to study rook placements avoiding sets of patterns.

Representations of the Rook-Brauer Algebra

We study the representation theory of the rook-Brauer algebra , which has a of Brauer diagrams that allow for the possibility of missing edges. The Brauer, Temperley–Lieb, Motzkin, rook monoid, and symmetric group algebras are subalgebras of . We prove that is the centralizer of the orthogonal group on tensor space and that and are in Schur–Weyl duality. When x ∈ ℂ is chosen so that is semisimple, we use its Bratteli diagram to explicitly construct a complete set of irreducible representations for .

Gelfand Models for Diagram Algebras

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.

Conjugacy classes of Renner monoids

In this paper we describe conjugacy classes of a Renner monoid R with unit group W, the Weyl group. We show that every element in R is conjugate to an element ue where u∈W and e is an idempotent in a cross section lattice. Denote by W(e) and W⁎(e) the centralizer and stabilizer of e∈Λ in W, respectively. Let W(e) act by conjugation on the set of left cosets of W⁎(e) in W. We find that ue and ve (u,v∈W) are conjugate if and only if uW⁎(e) and vW⁎(e) are in the same orbit. As consequences, there is a one-to-one correspondence between the conjugacy classes of R and the orbits of this action. We then obtain a formula for calculating the number of conjugacy classes of R, and describe in detail the conjugacy classes of the Renner monoid of some J-irreducible monoids.We then generalize Munn conjugacy on a rook monoid to any Renner monoid and show that Munn conjugacy coincides with semigroup conjugacy, action conjugacy, and character conjugacy. We also show that the number of inequivalent irreducible representations of R over an algebraically closed field of characteristic zero equals the number of Munn conjugacy classes in R.