Diagram categories and invariant theory for classical groups and supergroups

. We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and supergroups, with some indications concerning extensions to quantum groups and quantum supergroups. Tensor functors from various diagram categories to categories of representations are introduced and their properties are investigated, leading to first and second fundamental theorems (FFT and SFT) of invariant theory for classical supergroups, which include the FFTs and SFTs of the classical groups as special cases. Application of diagrammatic methods enables the construction of a presentation for endomorphism algebras for the orthogonal and symplectic groups, leading to the solution of a problem raised by the work of Brauer and Weyl.

[1]  G. Lehrer,et al.  First fundamental theorems of invariant theory for quantum supergroups , 2019, European Journal of Mathematics.

[2]  K. Coulembier,et al.  Borelic pairs for stratified algebras , 2016, Advances in Mathematics.

[3]  G. Lehrer,et al.  THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP , 2014, Nagoya Mathematical Journal.

[4]  G. Lehrer,et al.  EQUIVALENCE OF A TANGLE CATEGORY AND A CATEGORY OF INFINITE DIMENSIONAL Uqpsl2q-MODULES , 2019 .

[5]  K. Coulembier.  Tensor ideals, Deligne categories and invariant theory , 2017, Selecta Mathematica.

[6]  Dominic R. Verity,et al.  ∞-Categories for the Working Mathematician , 2018 .

[7]  G. Lehrer,et al.  Invariants of the special orthogonal group and an enhanced Brauer category , 2016, 1612.03998.

[8]  G. Lehrer,et al.  The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup , 2016, 1602.04885.

[9]  G. Lehrer,et al.  Invariants of the orthosymplectic Lie superalgebra and super Pfaffians , 2015, 1507.01329.

[10]  C. Stroppel,et al.  Koszul gradings on Brauer algebras , 2015, 1504.03924.

[11]  H. H. Andersen,et al.  Cellularity of certain quantum endomorphism algebras , 2013, 1303.0984.

[12]  R. B. Zhang,et al.  The Brauer Category and Invariant Theory , 2012, 1207.5889.

[13]  J. Comes.  DELIGNE’S CATEGORY Rep(GLδ) AND REPRESENTATIONS OF GENERAL LINEAR SUPERGROUPS , 2012 .

[14]  C. Stroppel,et al.  Gradings on walled Brauer algebras and Khovanov’s arc algebra , 2011, 1107.0999.

[15]  G. Lehrer,et al.  THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOGONAL GROUP , 2011, 1102.3221.

[16]  P. Martin.  On Diagram Categories , Representation Theory and Statistical Mechanics , 2011 .

[17]  Jun Hu,et al.  On tensor spaces for Birman–Murakami–Wenzl algebras , 2010 .

[18]  行者 明彦.  Representation theory of algebraic groups and quantum groups , 2010 .

[19]  R. Wilson.  The classical groups , 2009 .

[20]  G. Lehrer,et al.  On Endomorphisms of Quantum Tensor Space , 2008, 0806.3807.

[21]  S. Parvathi,et al.  Noncommutative rings, group rings, diagram algebras and their applications : international conference, December 18-22, 2006, University of Madras, Chennai, India , 2008 .

[22]  G. Lehrer,et al.  Strongly multiplicity free modules for Lie algebras and quantum groups , 2006 .

[23]  Mei Si,et al.  A criterion on the semisimple Brauer algebras II , 2006, J. Comb. Theory A.

[24]  G. Lehrer,et al.  Cellular algebras and diagram algebras in representation theory , 2004 .

[25]  G. Lehrer,et al.  Diagram algebras, Hecke algebras and decomposition numbers at roots of unity , 2003 .

[26]  P. Etingof,et al.  Lectures on Quantum Groups , 2001 .

[27]  Leonard L. Scott,et al.  Quantum Weyl Reciprocity and Tilting Modules , 1998 .

[28]  R. Goodman,et al.  Representations and Invariants of the Classical Groups , 1998 .

[29]  P. Podlés,et al.  Introduction to Quantum Groups , 1997, q-alg/9704002.

[30]  R. B. Zhang,et al.  Structure and Representations of the Quantum General Linear Supergroup , 1996, q-alg/9611024.

[31]  G. Lehrer,et al.  Cellular algebras , 1996 .

[32]  R. B. Zhang,et al.  QUANTUM SUPERGROUPS AND TOPOLOGICAL INVARIANTS OF THREE-MANIFOLDS , 1994, hep-th/9408177.

[33]  V. Turaev.  Quantum Invariants of Knots and 3-Manifolds , 1994, hep-th/9409028.

[34]  T. Kerler.  On braided tensor categories , 1994, hep-th/9402018.

[35]  Nicolai Reshetikhin,et al.  Quantum Groups , 1993 .

[36]  Vladimir Turaev,et al.  Invariants of 3-manifolds via link polynomials and quantum groups , 1991 .

[37]  A. Bracken,et al.  Quantum group invariants and link polynomials , 1991 .

[38]  A. Bracken,et al.  Quantum Supergroups and Solutions of the {Yang-Baxter} Equation , 1990 .

[39]  V. Turaev.  OPERATOR INVARIANTS OF TANGLES, AND R-MATRICES , 1990 .

[40]  V. Turaev,et al.  Ribbon graphs and their invaraints derived from quantum groups , 1990 .

[41]  David N. Yetter,et al.  Braided Compact Closed Categories with Applications to Low Dimensional Topology , 1989 .

[42]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[43]  Joan S. Birman,et al.  Braids, link polynomials and a new algebra , 1989 .

[44]  Hans Wenzl,et al.  On the structure of Brauer’s centralizer algebras , 1988 .

[45]  V. Jones.  Hecke algebra representations of braid groups and link polynomials , 1987 .

[46]  Amitai Regev,et al.  Hook young diagrams with applications to combinatorics and to representations of Lie superalgebras , 1987 .

[47]  H. Weyl.  The Classical Groups , 1940 .

[48]  Richard Brauer,et al.  On Algebras Which are Connected with the Semisimple Continuous Groups , 1937 .