GRASSMANNIAN SEMIGROUPS AND THEIR REPRESENTATIONS

2018 ◽  
Vol 104 (3) ◽  
pp. 308-337
Author(s):  
VICTOR CAMILLO ◽  
MIODRAG C. IOVANOV
Keyword(s):  
Representation Theory ◽  
Partial Order ◽  
Finite Semigroup ◽  
Arbitrary Field ◽  
Young Diagrams ◽  
Internal Properties ◽  
Reduced Matrices ◽  
Multiplicative Structures

The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the $\text{Gl}_{n}(\mathbb{K})$ action on the $n\times n$ matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in $M_{n}(\mathbb{K})$. We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.

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.

scholarly journals Hook representations of the symmetric groups

1971 ◽  
Vol 12 (2) ◽  
pp. 136-149 ◽  
Author(s):  
M. H. Peel
Keyword(s):  
Representation Theory ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Modular Representation ◽  
Splitting Field ◽  
Arbitrary Field ◽  
Modular Representation Theory ◽  
Prime Fields

In this paper we are concerned with the representation theory of the symmetric groupsover a field K of characteristic p. Every field is a splitting field for the symmetric groups. Consequently, in order to study the modular representation theory of these groups, it is sufficient to work over the prime fields. However, we take K to be an arbitrary field of characteristic p, since the presentation of the results is not affected by this choice. Sn denotes the group of permutations of {x1, …, xn], where x1,…,xn are independent indeterminates over K. The group algebra of Sn with coefficients in K is denoted by Фn.

Some combinatorial results involving Young diagrams

1978 ◽  
Vol 83 (1) ◽  
pp. 1-10 ◽  
Author(s):  
G. D. James
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Symmetric Group ◽  
Young Diagram ◽  
New Method ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Sufficient Criterion ◽  
Power Diagram ◽  
Necessary And Sufficient

In the first half of this paper we introduce a new method of examining the q-hook structure of a Young diagram, and use it to prove most of the standard results about q-cores and q-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q-weight and says how many of them are q-regular. In the case where q is prime, this tells us how many ordinary and q-modular irreducible representations of the symmetric group there are in a given q-block. None of the results of section 2 is original. In the next section we give a new definition, the p-power diagram, which is closely connected with the p-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be p-regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p. In this paper we derive combinatorial results involving the p-power diagram, and in a later article we investigate the relevant representation theory.

scholarly journals k-stretchability of entanglement, and the duality of k-separability and k-producibility

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 204 ◽  
Author(s):  
Szilárd Szalay
Keyword(s):  
Detailed Analysis ◽  
Partial Order ◽  
Classification Scheme ◽  
Quantum Systems ◽  
General Treatment ◽  
Young Diagrams ◽  
Multipartite Quantum Systems ◽  
Maximal Size ◽  
Correlation Properties

The notions of k-separability and k-producibility are useful and expressive tools for the characterization of entanglement in multipartite quantum systems, when a more detailed analysis would be infeasible or simply needless. In this work we reveal a partial duality between them, which is valid also for their correlation counterparts. This duality can be seen from a much wider perspective, when we consider the entanglement and correlation properties which are invariant under the permutations of the subsystems. These properties are labeled by Young diagrams, which we endow with a refinement-like partial order, to build up their classification scheme. This general treatment reveals a new property, which we call k-stretchability, being sensitive in a balanced way to both the maximal size of correlated (or entangled) subsystems and the minimal number of subsystems uncorrelated with (or separable from) one another.

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