Access the full text.
Sign up today, get DeepDyve free for 14 days.
A. Molev (1999)
Weight bases of Gelfand-Tsetlin type for representations of classical Lie algebrasJournal of Physics A, 33
B. LakshmiBai, C. Musili, C. Seshadri (1979)
Geometry of $G/P$Bulletin of the American Mathematical Society, 1
Many of the weight bases that can be realized by combinatorial methods similar to those employed here enjoy certain of the 'extremal' properties first studied in
P. Hersh, Cristian Lenart (2008)
Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin BasisElectron. J. Comb., 17
R. Green (2013)
Combinatorics of Minuscule Representations
R. Donnelly, Scott Lewis, Robert Pervine (2003)
Constructions of representations of o(2n+1, C) that imply Molev and Reiner-Stanton lattices are strongly SpernerDiscret. Math., 263
R. Donnelly, Scott Lewis, Robert Pervine (2006)
Solitary and edge-minimal bases for representations of the simple lie algebra G2Discret. Math., 306
A Weyl group generalization of skew Schur functions
W. Fulton, J. Harris (1991)
Representation Theory: A First Course
R. Donnelly, Molly Dunkum (2020)
Gelfand-Tsetlin-type weight bases for all special linear Lie algebra representations corresponding to skew Schur functionsAdv. Appl. Math., 139
A. Molev (1998)
A Basis for Representations of Symplectic Lie AlgebrasCommunications in Mathematical Physics, 201
(1978)
Seshadri
Katheryn Beck (2018)
Distributive lattice models of the type B elementary Weyl group symmetric functions
A. Molev (2002)
Gelfand-Tsetlin bases for classical Lie algebrasarXiv: Representation Theory
Ararat Harutyunyan, Lucas Pastor, Stéphan Thomassé (2015)
Disproving the normal graph conjectureJ. Comb. Theory, Ser. B, 147
A. Molev (1999)
A Weight Basis for Representations of Even Orthogonal Lie Algebras
(1999)
Molev
M Aigner (1979)
10.1007/978-1-4615-6666-3Combinatorial Theory
Robert Proctor (1990)
Solution of a sperner conjecture of stanley with a construction of gelfandJ. Comb. Theory, Ser. A, 54
Mireille Bousquet-Mélou, Guillaume Chapuy, Michael Drmota, Sergi Elizalde (2023)
Enumerative CombinatoricsOberwolfach Reports
R. Donnelly (2003)
Extremal Properties of Bases for Representations of Semisimple Lie AlgebrasJournal of Algebraic Combinatorics, 17
(2000)
On the work of E
(1988)
Tsetlin
EB Dynkin (1950)
Certain properties of the system of weights of linear representations of semisimple Lie groupsDokl. Acad. Nauk SSSR, 71
R. Donnelly (2018)
Finite diamond-colored modular and distributive lattices with applications to combinatorial Lie representation theory
J. Humphreys (1973)
Introduction to Lie Algebras and Representation Theory
(1962)
Lie Algebras (A corrected republication of the work originally
Robert Proctor (1984)
Bruhat Lattices, Plane Partition Generating Functions, and Minuscule RepresentationsEur. J. Comb., 5
R. Stanley (1980)
Weyl Groups, the Hard Lefschetz Theorem, and the Sperner PropertySIAM J. Algebraic Discret. Methods, 1
R. Donnelly (2018)
Poset models for Weyl group analogs of symmetric functions and Schur functionsarXiv: Combinatorics
J. Lepowsky (1980)
Application of the numerator formula to k-rowed plane partitionsAdvances in Mathematics, 35
E. Dynkin, A. I︠U︡shkevich, G. Seitz, A. Onishchik (2000)
Selected papers of E.B. Dynkin with commentary
Robert Proctor (1982)
Representations of $\mathfrak{sl}( 2,\mathbb{C} )$ on Posets and the Sperner PropertySiam Journal on Algebraic and Discrete Methods, 3
I. Gelfand, Michael Zetlin (1950)
Finite-dimensional representations of the group of unimodular matrices, 71
We construct every finite-dimensional irreducible representation of the simple Lie algebra of type E7\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textsf {E}}_{7}$$\end{document} whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type E7\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textsf {E}}_{7}$$\end{document} root system. As a consequence, we obtain constructions of each finite-dimensional irreducible representation of the simple Lie algebra of type E6\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textsf {E}}_{6}$$\end{document} whose highest weight is a nonnegative integer linear combination of the two dominant minuscule E6\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textsf {E}}_{6}$$\end{document}-weights. Our constructions are explicit in the sense that, if the representing space is d-dimensional, then a weight basis is provided such that all entries of the d×d\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d \times d$$\end{document} representing matrices of the Chevalley generators are obtained via explicit, non-recursive formulas. To effect this work, we introduce what we call E6\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textsf {E}}_{6}$$\end{document}- and E7\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textsf {E}}_{7}$$\end{document}-polyminuscule lattices that analogize certain lattices associated with the famous special linear Lie algebra representation constructions obtained by Gelfand and Tsetlin.
Applicable Algebra in Engineering Communication and Computing – Springer Journals
Published: May 26, 2023
Keywords: Simple Lie algebra representation; Root system; Minuscule weight; Weyl group; Weyl symmetric function; Ranked poset; Diamond-colored distributive lattice; Splitting poset; Weight basis supporting graph/representation diagram; 17B10 (22E70, 05E10)
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.