scholarly journals Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis

10.37236/305 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Patricia Hersh ◽  
Cristian Lenart
Keyword(s):  
Lie Algebras ◽  
Simple Algorithm ◽  
Irreducible Representations ◽  
Combinatorial Proof ◽  
Young Tableaux ◽  
Constructive Approach ◽  
Function Identity ◽  
New Interpretation ◽  
Combinatorial Constructions ◽  
Extremal Properties

This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of $\mathfrak{ sl}$$_n$, the celebrated Gelfand-Tsetlin basis is the only such basis known. Using the setup of supporting graphs developed by Donnelly, we present a new interpretation and a simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rational function identity (all the known proofs use more sophisticated algebraic tools). A constructive approach to the Gelfand-Tsetlin formulas is then given, based on a simple algorithm for solving certain equations on the lattice of semistandard Young tableaux. This algorithm also implies certain extremal properties of the Gelfand-Tsetlin basis.

scholarly journals Combinatorics of Generalized Exponents

2018 ◽  
Vol 2020 (16) ◽  
pp. 4942-4992 ◽  
Author(s):  
Cédric Lecouvey ◽  
Cristian Lenart
Keyword(s):  
Root System ◽  
Lie Algebras ◽  
Finite Type ◽  
Irreducible Representations ◽  
Combinatorial Proof ◽  
Combinatorial Formula ◽  
Branching Rules ◽  
Crystal Graphs ◽  
Type C ◽  
Generalized Exponents

Abstract We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type $A_{n-1}$, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type $C_{n}$, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type $A_{2n-1}$, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig $t$-analogs associated to zero-weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula and discuss some implications to relating two type $C$ branching rules. Our methods are expected to extend to the orthogonal types.

scholarly journals Constructions of Representations of Rank Two Semisimple Lie Algebras with Distributive Lattices

10.37236/1135 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
L. Wyatt Alverson II ◽  
Robert G. Donnelly ◽  
Scott J. Lewis ◽  
Robert Pervine
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Representations ◽  
Distributive Lattices ◽  
Semisimple Lie Algebra ◽  
Order Ideals ◽  
Symplectic Representations ◽  
Extremal Properties

We associate one or two posets (which we call "semistandard posets") to any given irreducible representation of a rank two semisimple Lie algebra over ${\Bbb C}$. Elsewhere we have shown how the distributive lattices of order ideals taken from semistandard posets (we call these "semistandard lattices") can be used to obtain certain information about these irreducible representations. Here we show that some of these semistandard lattices can be used to present explicit actions of Lie algebra generators on weight bases (Theorem 5.1), which implies these particular semistandard lattices are supporting graphs. Our descriptions of these actions are explicit in the sense that relative to the bases obtained, the entries for the representing matrices of certain Lie algebra generators are rational coefficients we assign in pairs to the lattice edges. In Theorem 4.4 we show that if such coefficients can be assigned to the edges, then the assignment is unique up to products; we conclude that the associated weight bases enjoy certain uniqueness and extremal properties (the "solitary" and "edge-minimal" properties respectively). Our proof of this result is uniform and combinatorial in that it depends only on certain properties possessed by all semistandard posets. For certain families of semistandard lattices some of these results were obtained in previous papers; in Proposition 5.6 we explicitly construct new weight bases for a certain family of rank two symplectic representations. These results are used to help obtain in Theorem 5.1 the classification of those semistandard lattices which are supporting graphs.

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

scholarly journals Specht modules for finite reflection groups

1995 ◽  
Vol 37 (3) ◽  
pp. 279-287 ◽  
Author(s):  
S. HalicioǦlu
Keyword(s):  
Present Author ◽  
Symmetric Groups ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Arbitrary Field ◽  
Young Tableaux ◽  
Reflection Groups ◽  
Specht Modules ◽  
Irreducible Modules ◽  
Original Approach

Over fields of characteristic zero, there are well known constructions of the irreducible representations, due to A. Young, and of irreducible modules, called Specht modules, due to W. Specht, for the symmetric groups Sn which are based on elegant combinatorial concepts connected with Young tableaux etc. (see, e.g. [13]). James [12] extended these ideas to construct irreducible representations and modules over an arbitrary field. Al-Aamily, Morris and Peel [1] showed how this construction could be extended to cover the Weyl groups of type Bn. In [14] Morris described a possible extension of James' work for Weyl groups in general. Later, the present author and Morris [8] gave an alternative generalisation of James' work which is an extended improvement and extension of the original approach suggested by Morris. We now give a possible extension of James' work for finite reflection groups in general.

scholarly journals On Real Irreducible Representations of Lie Algebras

1959 ◽  
Vol 14 ◽  
pp. 59-83 ◽  
Author(s):  
Nagayoshi Iwahori
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Number Field ◽  
Irreducible Representations ◽  
The Real ◽  
Representations Of Lie Algebras ◽  
Real Matrices ◽  
Real Number Field

Let us consider the following two problems:Problem A. Let g be a given Lie algebra over the real number field R. Then find all real, irreducible representations of g.Problem B. Let n be a given positive integer. Then find all irreducible subalgebras of the Lie algebra ôí(w, R) of all real matrices of degree n.

scholarly journals TWO PERMANENTS IN THE UNIVERSAL ENVELOPING ALGEBRAS OF THE SYMPLECTIC LIE ALGEBRAS

2009 ◽  
Vol 20 (03) ◽  
pp. 339-368 ◽  
Author(s):  
MINORU ITOH
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Universal Enveloping Algebra ◽  
Irreducible Representations ◽  
General Linear ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Central Elements ◽  
Orthogonal Lie Algebra

This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra.

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