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 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 Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

On the Uniqueness of the Cyclic Group of Order n

1992 ◽  
Vol 99 (6) ◽  
pp. 545-547 ◽  
Author(s):  
Dieter Jungnickel
Keyword(s):  
Cyclic Group

scholarly journals A note on frame starters

1996 ◽  
Vol 53 (2) ◽  
pp. 293-297
Author(s):  
Cheng-De Wang
Keyword(s):  
Cyclic Group ◽  
Unique Element ◽  
Left Frame

We construct frame starters in Z2n − {0, n}, for n ≡ 0, 1 mod 4, where Z2n denotes the cyclic group of order 2n. We also construct left frame starters in Q2n − {e, αn}, where Q2n is the dicyclic group of order 4n and αn is the unique element of order 2 in Q2n.

Groups with a Cyclic Group as Lattice-Homomorph

1948 ◽  
Vol 49 (2) ◽  
pp. 347
Author(s):  
Philip M. Whitman
Keyword(s):  
Cyclic Group

scholarly journals Semigroups with few endomorphisms

1969 ◽  
Vol 10 (1-2) ◽  
pp. 162-168 ◽  
Author(s):  
Vlastimil Dlab ◽  
B. H. Neumann
Keyword(s):  
Cyclic Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Infinite Groups ◽  
Infinite Cyclic Group

Large finite groups have large automorphism groups [4]; infinite groups may, like the infinite cyclic group, have finite automorphism groups, but their endomorphism semigroups are infinite (see Baer [1, p. 530] or [2, p. 68]). We show in this paper that the corresponding propositions for semigroups are false.

scholarly journals The solution of length three equations over groups

1983 ◽  
Vol 26 (1) ◽  
pp. 89-96 ◽  
Author(s):  
James Howie
Keyword(s):  
Free Product ◽  
Cyclic Group ◽  
Normal Closure ◽  
Infinite Cyclic Group ◽  
Canonical Map ◽  
Equations Over Groups

Let G be a group, and let r = r(t) be an element of the free product G * 〈G〉 of G with the infinite cyclic group generated by t. We say that the equation r(t) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * 〈G〉 to G with r in its kernel. We say that r(t) = 1 has a solution over G if G can be embedded in a group H such that r(t) = 1 has a solution in H. This property is equivalent to the canonical map from G to 〈G, t|r〉 (the quotient of G * 〈G〉 by the normal closure of r) being injective.

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