## Combinatorial representation theory (1999)

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Venue: | in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–97), MSRI Publ. 38 |

Citations: | 13 - 0 self |

### BibTeX

@INPROCEEDINGS{Barcelo99combinatorialrepresentation,

author = {Hélène Barcelo and Arun Ram},

title = {Combinatorial representation theory},

booktitle = {in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–97), MSRI Publ. 38},

year = {1999},

pages = {23--90},

publisher = {University Press}

}

### OpenURL

### Abstract

Abstract. We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, �), and later for certain generalizations, when known. Background material and more specialized results are given in a series of appendices. We give a personal view of the field while remaining aware that there is much important and beautiful work that we have been unable to mention.

### Citations

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Introduction to Lie Algebras and Representation Theory
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Citation Context ...rticularly elusive theorem. Sometimes these representations lead to a completely new understanding of previously known facts. A famous example (which unfortunately we won’t have space to discuss, see =-=[Hu1]-=-) is the Verma6 HÉLÈNE BARCELO AND ARUN RAM module, which was discovered in the mid 1960s and completely changed representation theory. M. The modular case In the modular case we have the following i... |

728 |
A guide to quantum groups
- Chari, Pressley
- 1994
(Show Context)
Citation Context ... Birman and Wenzl [BW] and Murakami [Mu1]. It was realized early [Re] [Wz3] that these arise as tensor power centralizers but there was no proof in the literature for some time. See the references in =-=[CP]-=- §10.2. I. What are the irreducibles? Indexing of the representations of tensor power centralizer algebras follows from double centralizer theory (see Weyl [Wy1]) and a good understanding of the index... |

453 |
Reflection groups and Coxeter groups, Cambridge
- Humphreys
- 1990
(Show Context)
Citation Context ...nite Coxeter groups of types An−1, Bn, Dn, E6, E7, E8, F4, G2 = I2(6). References The most comprehensive reference for finite groups generated by reflections is [Bou1]. See also the book of Humphreys =-=[Hu2]-=-.COMBINATORIAL REPRESENTATION THEORY 45 B2. Complex reflection groups A complex reflection is an invertible linear transformation of C n of finite order which has exactly one eigenvalue that is not 1... |

400 |
Groupes et Algèbres de Lie
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(Show Context)
Citation Context ...ible Weyl groups are the irreducible finite Coxeter groups of types An−1, Bn, Dn, E6, E7, E8, F4, G2 = I2(6). References The most comprehensive reference for finite groups generated by reflections is =-=[Bou1]-=-. See also the book of Humphreys [Hu2].COMBINATORIAL REPRESENTATION THEORY 45 B2. Complex reflection groups A complex reflection is an invertible linear transformation of C n of finite order which ha... |

378 | Representation Theory of Finite Groups and Associative Algebras - Curtis, Reiner - 1962 |

373 |
Representation theory, A first course
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- 1991
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Citation Context ...xing of irreducible representations given in (Ia) is due to Cartan and Killing, the founders of the theory, from around the turn of the century. Introductory treatments of this result can be found in =-=[FH]-=- and [Hu1].COMBINATORIAL REPRESENTATION THEORY 19 (2) The first equality in (Ib) is due to Littelmann [Li1], but his later article [Li2] has some improvements and can be read independently, so we rec... |

368 |
Linear Representations of Finite Groups
- Serre
- 1977
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Citation Context ...o write M = ∑ λ cλV λ has been given a formal setting which is called the Grothendieck ring. In other words, the formal object which allows us to write such identities has been defined carefully. See =-=[Se1]-=- for precise definitions of the Grothendieck ring. Answers should be of the form . . . Now we come to the adjective “Combinatorial.” It refers to the way in which we give the answers to the main quest... |

365 |
Symmetric functions and Hall polynomials, 2nd edition
- Macdonald
(Show Context)
Citation Context ...etimes a representation is exactly what is most helpful for solving a combinatorial problem. One example of this is in the recent solution of the the last few plane partition conjectures. See [Sta1], =-=[Mac]-=- I §5 Ex. 13-18, for the statement of the problem and [Ku13] and [Ste5-7] for the solutions. These solutions were motivated by the method of Proctor [Prc]. The main point of all this is that a combina... |

359 |
G Lusztig, Representations of Coxeter groups and Hecke algebras
- Kazhdan
- 1979
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Citation Context ...x A3 for more information in the symmetric group case. It is possible that this construction may be combinatorialized in the future, but to date no one has done this. The Kazhdan-Lusztig construction =-=[KL1]-=- is a construction of certain representations called cell representations and it works for all finite Coxeter groups. The cell representations are almost irreducible but unfortunately not irreducible ... |

343 |
Lie groups, Lie algebras, and Their Representations". Graduate Texts in Mathematics 102
- Varadarajan
- 1974
(Show Context)
Citation Context ...column strict tableaux. The second equality in (Ic) is the celebrated Weyl character formula which was originally proved in [Wy2]. A modern treatment of this formula can be found in [BtD], [Hu1], and =-=[Va]-=-. (5) The general restriction formula (S1) is due to Littelmann [Li2]. This is an analogue of the rule given in (S1) of the GL(n, C) results. In this case the formula is as a sum over paths which sati... |

242 |
Representation Theory and Complex Geometry
- Chriss, Ginzburg
- 1997
(Show Context)
Citation Context ...usztig construction. (11) Springer’s construction is a geometric construction. In this construction the irreducible module S λ is realized as the top cohomology group of a certain variety, see [Spr], =-=[CG]-=-, and Appendix A3. (12) There are many ways of constructing new representations from old ones. Among the common techniques are restriction, induction, and tensoring. The special representations (S1), ... |

241 |
Finite groups of Lie type: Conjugacy classes and complex characters
- Carter
- 1993
(Show Context)
Citation Context ...Ariki-Koike [AK] and Ariki [Ari] say that the “Hecke algebras” of G(r, p, n) are q-deformations of the group algebras of the groups G(r, p, n). Thus, it follows from the Tits deformation theorem (see =-=[Ca]-=- Chapt 10, 11.2 and [CR2] §68.17) that the indexings and dimension formulas for the irreducible representations of these algebras must be the same as the indexings and dimension formulas for the group... |

228 |
Infinite dimensional Lie algebras, Third edition
- Kac
- 1990
(Show Context)
Citation Context ...ew of them have been stated or interpreted through a combinatorialists eyes. The world is a gold mine, yet to be mined! Notes and references (1) An introductory reference to Kac-Moody Lie algebras is =-=[Kc]-=- . This book contains a good description of the basic representation theory of these algebras. We don’t know of a good introductory reference for the Kac-Moody groups case, we would suggest beginning ... |

213 |
Duality for representations of a reductive group over a finite field
- Deligne, Lusztig
- 1982
(Show Context)
Citation Context ...has been a concerted effort to extend these results to other finite Chevalley groups. G. Lusztig [Lu8-11] has made important contributions to this field; in particular, the results of Deligne-Lusztig =-=[DL]-=- are fundamental. However, this is a geometric approach rather than a combinatorial one and there is much work to be done for combinatorialists, even in interpreting the known results from combinatori... |

199 |
Finite unitary reflection groups
- Shephard, Todd
- 1954
(Show Context)
Citation Context ...d which have exactly one eigenvalue that is not 1. Every finite Coxeter group is also a finite complex reflection group. The finite complex reflection groups have been classified by Shephard and Todd =-=[ST]-=- and each such group is one of the groups (a) G(r, p, n), where r, p, n are positive integers such that p divides r, or (b) one of 34 “exceptional” finite complex reflection groups. The groups G(r, p,... |

176 |
Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts
- Borel
(Show Context)
Citation Context ...f the irreducible modules of Sn is the Springer construction. References See [Mac] II §3 Ex. 1 for a description of the variety Bu and its structure. The theorem of Borel stated in (A3.1) is given in =-=[Bo]-=- and [BGG]. The references quoted in the text above will provide a good introduction to the Springer theory. The beautiful combinatorics of Springer theory has been studied by Barcelo [Ba], Garsia-Pro... |

171 | Representation Theory. A first course,” Graduate Texts - Fulton, Harris - 1991 |

167 | State models and the Jones polynomial, Topology 26 - Kauffman - 1987 |

164 |
tom Dieck T., Representations of compact Lie groups, Graduate Texts
- Bröcker
- 1985
(Show Context)
Citation Context ...a weighted sum of column strict tableaux. The second equality in (Ic) is the celebrated Weyl character formula which was originally proved in [Wy2]. A modern treatment of this formula can be found in =-=[BtD]-=-, [Hu1], and [Va]. (5) The general restriction formula (S1) is due to Littelmann [Li2]. This is an analogue of the rule given in (S1) of the GL(n, C) results. In this case the formula is as a sum over... |

161 |
On algebras which are connected with the semisimple continuous groups
- Brauer
- 1937
(Show Context)
Citation Context ...ebras in 1964 in connection with GL(n, Fq). Jimbo [Ji] realized that they arise as tensor power centralizer algebras for quantum groups. Brauer algebras. These algebras were defined by Brauer in 1937 =-=[Br]-=-. Brauer also proved that they are tensor power centralizers. Birman-Murakami-Wenzl algebras. These algebras are due to Birman and Wenzl [BW] and Murakami [Mu1]. It was realized early [Re] [Wz3] that ... |

154 |
A q-analog of U(gln+1), Hecke algebra, and the Yang-Baxter equation
- Jimbo
- 1986
(Show Context)
Citation Context ...l in making them so important in combinatorial representation theory today. The Iwahori-Hecke algebras of type An−1. Iwahori [Iw] introduced these algebras in 1964 in connection with GL(n, Fq). Jimbo =-=[Ji]-=- realized that they arise as tensor power centralizer algebras for quantum groups. Brauer algebras. These algebras were defined by Brauer in 1937 [Br]. Brauer also proved that they are tensor power ce... |

154 | Affine Hecke algebras and their graded version - Lusztig - 1989 |

149 |
Representations of Hecke algebras of general linear groups
- Dipper, James
- 1986
(Show Context)
Citation Context ...ne appropriate choice for the analogue of Young symmetrizers for Hecke algebras. The definitions in the literature are due to Gyoja [Gy], for the Iwahori-Hecke algebras of type An−1, Dipper and James =-=[DJ1]-=- and Murphy [M1-2] for the Iwahori-Hecke algebras of type An−1, King and Wybourne [KW] and Duchamp, et al [DK] for the Iwahori-Hecke algebras of type An−1, Dipper, James, and Murphy [DJ2], [DJM] for t... |

149 | Methods of Representation Theory - Curtis, Reiner - 1981 |

132 |
Coxeter graphs and towers of algebras
- Goodman, Harpe, et al.
- 1989
(Show Context)
Citation Context ...dexings and dimension formulas for the irreducible representations are as follows: Temperley-Lieb algebras. These results are classical and can be found in the book by Goodman, de la Harpe, and Jones =-=[GHJ]-=-.COMBINATORIAL REPRESENTATION THEORY 25 Brauer algebras. These results were known to Brauer [Br] and Weyl [Wy]. An important combinatorial point of view was given by Berele [Be1-2] and further develo... |

131 |
Index for subfactors, Invent
- Jones
- 1983
(Show Context)
Citation Context ...erley-Lieb algebras. These algebras are due, independently, to many different people. Some of the discoverers were Rumer-Teller-Weyl [RTW], Penrose [P1-2], Temperley-Lieb [TL], Kaufmann [Ka] and Jones=-=[Jo2]-=-. The work of V. Jones was crucial in making them so important in combinatorial representation theory today. The Iwahori-Hecke algebras of type An−1. Iwahori [Iw] introduced these algebras in 1964 in ... |

131 |
Crystalizing the q-Analogue of Universal Enveloping Algebras Commun
- Kashiwara
- 1990
(Show Context)
Citation Context ...ths Pπλ used in (Ib-c) is a modified description of the same set which appeared in Lakshmibai’s conjecture. Another important influence on Littelmann in his work was Kashiwara’s work on crystal bases =-=[Ksh]-=-. Part II 5. Generalizing the Sn results Having the above results for the symmetric group in hand we would like to try to generalize as many of the Sn results to other similar groups and algebras as w... |

126 |
Proof of the Deligne-Langlands conjecture for Hecke algebras
- Kazhdan, Lusztig
- 1987
(Show Context)
Citation Context ... Hecke algebras. The case of affine Coxeter groups was done by Kato [Kat] using Clifford theory ideas. The case of affine Hecke algebras has been intensely studied by Lusztig [Lu1-7], Kazhdan-Lusztig =-=[KL2]-=-, and Ginzburg [G], [CG], but most of this work is very geometric and relies on intersection cohomology/K-theory methods. Hopefully some of their work will be made combinatorial in the near future. (3... |

124 | Representation theory, Graduate Texts - Fulton, Harris - 1991 |

118 |
Real Reductive Groups
- Wallach
- 1988
(Show Context)
Citation Context ...binatorial representation theory for p-adic groups. (6) The best place to read about the representation theory of real reductive groups is in the books of D. Vogan and N. Wallach [AV], [Vg1] , [Vg2], =-=[Wa]-=-. (7) The Virasoro algebra is a Lie algebra that seems to turn up in every back alley of representation theory. One can only surmise that it must have a beautiful combinatorial representation theory t... |

115 |
Schubert cells and cohomology of the spaces
- Bernstein, Gel’fand, et al.
- 1973
(Show Context)
Citation Context ...educible modules of Sn is the Springer construction. References See [Mac] II §3 Ex. 1 for a description of the variety Bu and its structure. The theorem of Borel stated in (A3.1) is given in [Bo] and =-=[BGG]-=-. The references quoted in the text above will provide a good introduction to the Springer theory. The beautiful combinatorics of Springer theory has been studied by Barcelo [Ba], Garsia-Procesi [GP],... |

115 |
The theory of group characters and matrix representations of groups
- Littlewood
- 2006
(Show Context)
Citation Context ...s of the irreducibles as the number of column strict tableaux follows from the work of Kostka [Kk] and Schur [Sc1]. The “hookcontent” formula appears in [Mac] I §3 Ex. 4, where the book of Littlewood =-=[Lw]-=- is quoted.COMBINATORIAL REPRESENTATION THEORY 17 (5) The construction of the irreducibles by Young symmetrizers appeared in 1939 in the influential book [Wy1] of H. Weyl. It was generalized to the s... |

115 |
Hecke algebras of type An and subfactors
- Wenzl
- 1988
(Show Context)
Citation Context ...ralized to the case of G(r, p, n) and its “Hecke algebras”. C. How do we construct the irreducible modules? Analogues of Young’s seminormal representations have been given by Hoefsmit [Hfs] and Wenzl =-=[Wz1]-=-, independently, for Iwahori-Hecke algebras of type An−1, Hoefsmit [Hfs], for Iwahori-Hecke algebras of types Bn and Dn, Ariki and Koike [AK] for the “Hecke algebras” of G(r, 1, n), and Ariki [Ari] fo... |

109 | Angular momentum: an approach to combinatorial spacetime - Penrose - 1971 |

106 |
Lectures on quantum groups
- Jantzen
- 1996
(Show Context)
Citation Context ... case (3) are given in Section 4. The reduction of cases (1) and (2) to case (3) are outlined in [Se2], and given in more detail in [Va] and [BtD]. The reduction of (4) to (3) is given in [CP] and in =-=[Ja]-=-.28 HÉLÈNE BARCELO AND ARUN RAM Partial results for further generalizations Some partial results along the lines of the results (Ia-c) and (S1-2) for GL(n, C) and complex semisimple Lie algebras have... |

105 |
Paths and root operators in representation theory
- Littelmann
- 1995
(Show Context)
Citation Context ...roups and Lie algebras were laid in the fundamental work of Weyl [Wy2] in 1925, it is only recently that a complete generalization of the tableaux results for GL(n, C) has been obtained by Littelmann =-=[Li2]-=-. The results which we state below are generalizations of those given for GL(n, C) in the last section; partitions get replaced by points in a lattice called P + , and column strict tableaux get repla... |

104 |
Methods of Representation Theory with Applications to Finite Groups
- Curtis, Reiner
- 1981
(Show Context)
Citation Context ...e symmetric group Sn. Then one can use Clifford theory again to reduce the G(r, p, n) case to the G(r, 1, n) case. The original reference for Clifford theory is [Cl] and the book by Curtis and Reiner =-=[CR2]-=- has a modern treatment. The articles [Ste3] and [HR2] explain how the reduction from G(r, p, n) to G(r, 1, n) is done. The dimension and character theory for the case G(r, 1, n) has an excellent mode... |

98 |
A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras
- Littelmann
- 1994
(Show Context)
Citation Context ... from around the turn of the century. Introductory treatments of this result can be found in [FH] and [Hu1].COMBINATORIAL REPRESENTATION THEORY 19 (2) The first equality in (Ib) is due to Littelmann =-=[Li1]-=-, but his later article [Li2] has some improvements and can be read independently, so we recommend the later article. This formula for the dimension of the irreducible representation, the number of pa... |

96 |
An invariant of regular isotopy
- Kauffman
- 1990
(Show Context)
Citation Context ...3] contain further important information about the BMW-algebras. (2) Although the tangle description of the BMW algebra was always in everybody’s minds it was Kaufmann that really made it precise see =-=[Ka2]-=-. B8. The Temperley-Lieb algebras TLk(x) A TLk-diagram is a Brauer diagram on k dots which can be drawn with no crossings of edges. The Temperley-Lieb algebra TLk(x) is the subalgebra of the Brauer al... |

87 |
Relations between the percolation and colouring problem and other graphtheoretical problems associated with regular planar lattices: some exact results for the percolation problem
- Temperley, Lieb
- 1971
(Show Context)
Citation Context ...algebras are as follows. Temperley-Lieb algebras. These algebras are due, independently, to many different people. Some of the discoverers were Rumer-Teller-Weyl [RTW], Penrose [P1-2], Temperley-Lieb =-=[TL]-=-, Kaufmann [Ka] and Jones[Jo2]. The work of V. Jones was crucial in making them so important in combinatorial representation theory today. The Iwahori-Hecke algebras of type An−1. Iwahori [Iw] introdu... |

85 |
link polynomials and a new algebra
- Birman, Wenzl
- 1989
(Show Context)
Citation Context ...er algebras. These algebras were defined by Brauer in 1937 [Br]. Brauer also proved that they are tensor power centralizers. Birman-Murakami-Wenzl algebras. These algebras are due to Birman and Wenzl =-=[BW]-=- and Murakami [Mu1]. It was realized early [Re] [Wz3] that these arise as tensor power centralizers but there was no proof in the literature for some time. See the references in [CP] §10.2. I. What ar... |

81 | A Hecke algebra of (Z/rZ) ≀ Sn and construction of its irreducible representations - Ariki, Koike - 1994 |

79 |
Foncteurs analytiques et espèces de structures
- Joyal
- 1985
(Show Context)
Citation Context ...tion on the homology of the partition lattice which also, miraculously, appears as a representation on the free Lie algebra. We won’t have space to discuss this here, see the original references [Hn],=-=[Jy]-=-, [Kl], [Sta2], the article [Gar] for some further basics, and [Ba2] for a study of how it can be that this representation appears in two completely different places. (d) Are they useful?8 HÉLÈNE BAR... |

71 | Cohomological induction and unitary representations - Knapp, Vogan - 1995 |

70 |
Representations of finite groups of Lie type
- Digne, Michel
- 1991
(Show Context)
Citation Context ... there is much work to be done for combinatorialists, even in interpreting the known results from combinatorial viewpoint. A good introductory treatment of this theory is the book by Digne and Michel =-=[DM]-=-. The original work of Green is treated in [Mac] Chapt. IV.COMBINATORIAL REPRESENTATION THEORY 29 (5) The representation theory of p-adic Lie groups has been studied intensely by representation theor... |

70 |
The nil Hecke ring and cohomology of G/P for a Kac–Moody group
- Kostant, Kumar
- 1986
(Show Context)
Citation Context ...ntains a good description of the basic representation theory of these algebras. We don’t know of a good introductory reference for the Kac-Moody groups case, we would suggest beginning with the paper =-=[KK]-=- and following the references there. (2) The basic introductory reference for Yangians and their basic representation theory is [CP], Chapter 12. See also the references given there. (3) The best intr... |

70 |
On the structure of Brauer’s centralizer algebra
- Wenzl
- 1988
(Show Context)
Citation Context ...ponding results for the Brauer algebra. The indexings and dimension formulas for the Temperley-Lieb and Brauer algebras also follow easily by using the techniques of the Jones basic construction, see =-=[Wz2]-=- and [HR1]. The references for the irreducible characters of the various tensor power centralizer algebras are as follows: Temperley-Lieb algebras. Character formulas can be derived easily by using Jo... |

70 | Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures - Howe - 1992 |

69 |
Some aspects of group acting on finite posets
- Stanley
- 1982
(Show Context)
Citation Context ...homology of the partition lattice which also, miraculously, appears as a representation on the free Lie algebra. We won’t have space to discuss this here, see the original references [Hn],[Jy], [Kl], =-=[Sta2]-=-, the article [Gar] for some further basics, and [Ba2] for a study of how it can be that this representation appears in two completely different places. (d) Are they useful?8 HÉLÈNE BARCELO AND ARUN ... |

66 |
The invariant theory of n × n matrices
- Procesi
- 1976
(Show Context)
Citation Context ...published result of L. Solomon, see [So2].22 HÉLÈNE BARCELO AND ARUN RAM The wall algebras. These algebras were introduced in a nice combinatorial form in [BC] and in other forms in [Ko] and Procesi =-=[Pr]-=- and other older invariant theory works [Wy]. All of these works were related to tensor power centralizers and/or fundamental theorems of invariant theory. The q-wall algebras. These algebras were int... |