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 A Specht Module Analog for the Rook Monoid

10.37236/1619 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Cheryl Grood
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Algebraic Structure ◽  
Irreducible Representations ◽  
Specht Module ◽  
Specht Modules ◽  
Rook Monoid ◽  
Combinatorial Construction

The wealth of beautiful combinatorics that arise in the representation theory of the symmetric group is well-known. In this paper, we analyze the representations of a related algebraic structure called the rook monoid from a combinatorial angle. In particular, we give a combinatorial construction of the irreducible representations of the rook monoid. Since the rook monoid contains the symmetric group, it is perhaps not surprising that the construction outlined in this paper is very similar to the classic combinatorial construction of the irreducible $S_n$-representations: namely, the Specht modules.

scholarly journals RSK Insertion for Set Partitions and Diagram Algebras

10.37236/1881 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Tom Halverson ◽  
Tim Lewandowski
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Group Algebras ◽  
Jeu De Taquin ◽  
Set Partitions ◽  
Partition Algebra ◽  
Diagram Algebras ◽  
Rook Monoid ◽  
Planar Partition ◽  
Symmetric Group Algebras

We give combinatorial proofs of two identities from the representation theory of the partition algebra ${\Bbb C} A_k(n)$, $n \ge 2k$. The first is $n^k = \sum_\lambda f^\lambda m_k^\lambda$, where the sum is over partitions $\lambda$ of $n$, $f^\lambda$ is the number of standard tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of "vacillating tableaux" of shape $\lambda$ and length $2k$. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is $B(2k) = \sum_\lambda (m_k^\lambda)^2$, where $B(2k)$ is the number of set partitions of $\{1, \ldots, 2k\}$. We show that this insertion restricts to work for the diagram algebras which appear as subalgebras of the partition algebra: the Brauer, Temperley-Lieb, planar partition, rook monoid, planar rook monoid, and symmetric group algebras.

scholarly journals A $\lambda$-ring Frobenius Characteristic for $G\wr S_n$

10.37236/1809 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Anthony Mendes ◽  
Jeffrey Remmel ◽  
Jennifer Wagner
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Reproducing Kernel ◽  
Irreducible Representations ◽  
Inner Product ◽  
Irreducible Components ◽  
Lambda Ring

A $\lambda$-ring version of a Frobenius characteristic for groups of the form $G \wr S_n$ is given. Our methods provide natural analogs of classic results in the representation theory of the symmetric group. Included is a method decompose the Kronecker product of two irreducible representations of $G\wr S_n$ into its irreducible components along with generalizations of the Murnaghan-Nakayama rule, the Hall inner product, and the reproducing kernel for $G\wr S_n$.

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.

Entanglement classification in the noninteracting Fermi gas

2017 ◽  
Vol 15 (07) ◽  
pp. 1750055 ◽  
Author(s):  
M. A. Jafarizadeh ◽  
F. Eghbalifam ◽  
S. Nami ◽  
M. Yahyavi
Keyword(s):  
Density Matrix ◽  
Fermi Gas ◽  
Reduced Density Matrix ◽  
Irreducible Representations ◽  
Entanglement Measure ◽  
Projection Operators ◽  
Entanglement Witness ◽  
Gas Density ◽  
Weyl Duality ◽  
Reduced Density

In this paper, entanglement classification shared among the spins of localized fermions in the noninteracting Fermi gas is studied. It is proven that the Fermi gas density matrix is block diagonal on the basis of the projection operators to the irreducible representations of symmetric group [Formula: see text]. Every block of density matrix is in the form of the direct product of a matrix and identity matrix. Then it is useful to study entanglement in every block of density matrix separately. The basis of corresponding Hilbert space are identified from the Schur–Weyl duality theorem. Also, it can be shown that the symmetric part of the density matrix is fully separable. Then it has been shown that the entanglement measure which is introduced in Eltschka et al. [New J. Phys. 10, 043104 (2008)] and Guhne et al. [New J. Phys. 7, 229 (2005)], is zero for the even [Formula: see text] qubit Fermi gas density matrix. Then by focusing on three spin reduced density matrix, the entanglement classes have been investigated. In three qubit states there is an entanglement measure which is called 3-tangle. It can be shown that 3-tangle is zero for three qubit density matrix, but the density matrix is not biseparable for all possible values of its parameters and its eigenvectors are in the form of W-states. Then an entanglement witness for detecting non-separable state and an entanglement witness for detecting nonbiseparable states, have been introduced for three qubit density matrix by using convex optimization problem. Finally, the four spin reduced density matrix has been investigated by restricting the density matrix to the irreducible representations of [Formula: see text]. The restricted density matrix to the subspaces of the irreducible representations: [Formula: see text], [Formula: see text] and [Formula: see text] are denoted by [Formula: see text], [Formula: see text] and [Formula: see text], respectively. It has been shown that some highly entangled classes (by using the results of Miyake [Phys. Rev. A 67, 012108 (2003)] for entanglement classification) do not exist in the blocks of density matrix [Formula: see text] and [Formula: see text], so these classes do not exist in the total Fermi gas density matrix.

A Remark by Philip Hall

1958 ◽  
Vol 1 (1) ◽  
pp. 21-23 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Linear Group ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Linear Transformations ◽  
Modern Physics ◽  
The Right ◽  
The Relationship

The relationship between the representation theory of the full linear group GL(d) of all non-singular linear transformations of degree d over a field of characteristic zero and that of the symmetric group Sn goes back to Schur and has been expounded by Weyl in his classical groups, [4; cf also 2 and 3]. More and more, the significance of continuous groups for modern physics is being pressed on the attention of mathematicians, and it seems worth recording a remark made to the author by Philip Hall in Edmonton.As is well known, the irreducible representations of Sn are obtainable from the Young diagrams [λ]=[λ1, λ2 ,..., λr] consisting of λ1 nodes in the first row, λ2 in the second row, etc., where λ1≥λ2≥ ... ≥λr and Σ λi = n. If we denote the jth node in the ith row of [λ] by (i,j) then those nodes to the right of and below (i,j), constitute, along with the (i,j) node itself, the (i,j)-hook of length hij.

ROBINSON-SCHENSTED CORRESPONDENCE FOR THE G-VERTEX COLORED PARTITION ALGEBRA

2010 ◽  
Vol 03 (02) ◽  
pp. 369-385 ◽  
Author(s):  
A. Tamilselvi
Keyword(s):  
Cyclic Group ◽  
Partition Algebra ◽  
Partition Algebras

In this paper, we develop the Robinson-Schensted correspondence for the G-vertex colored partition algebras, which gives the bijection between the set of G-vertex colored partition diagrams Pk(n, G) and the pairs of [Formula: see text]-vacillating tableaux of shape λ, [Formula: see text], [Formula: see text] where Γk= {µ ⊢ m|0 ≤ m ≤ k}, Ψq= {λ | λ = (q - j, 1j), j = 0, 1,…, q - 1} and G is a cyclic group of order q. We also derive the Knuth relations for the G-vertex colored partition algebra by using the Robinson-Schensted correspondence for the [Formula: see text]-vacillating tableau of shape [Formula: see text].

scholarly journals Extensions of differential representations of SL2 and tori

2012 ◽  
Vol 12 (1) ◽  
pp. 199-224 ◽  
Author(s):  
Andrey Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Difference Equations ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Irreducible Representations ◽  
Direct Sums ◽  
Rational Representation ◽  
Linear Differential ◽  
Linear Differential Algebraic Groups

AbstractLinear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with ${\mathbf{SL} }_{2} $ and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of ${\mathbf{SL} }_{2} $. In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.

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