AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Weight Multiplicities and Young Tableaux Through Affine Crystals
About this Title
Jang Soo Kim, Kyu-Hwan Lee and Se-jin Oh
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 283, Number 1401
ISBNs: 978-1-4704-5994-9 (print); 978-1-4704-7402-7 (online)
DOI: https://doi.org/10.1090/memo/1401
Published electronically: January 20, 2023
Table of Contents
Chapters
- Introduction
- 1. Affine Kac–Moody algebras
- 2. Crystals and Young walls
- 3. Young tableaux and almost even tableaux
- 4. Lattice paths and triangular arrays
- 5. Dominant maximal weights
- 6. Weight multiplicities and (spin) rigid Young tableaux
- 7. Level $2$ weight multiplicities: Catalan and Pascal triangles
- 8. Level $3$ weight multiplicities: Motzkin and Riordan triangles
- 9. Some level $k$ weight multiplicities when $k\to \infty$: Bessel triangle
- 10. Standard Young tableaux with a fixed number of rows of odd length
Abstract
The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac–Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the $($spin$)$ rigid tableaux, and prove that they are equinumerous to the weight multiplicities of the highest weight modules under our consideration. These new classes of Young tableaux arise from crystal basis elements for dominant maximal weights of the integrable highest weight modules over affine Kac–Moody algebras. By applying combinatorics of tableaux such as the Robinson–Schensted algorithm and new insertion schemes, and using integrals over orthogonal groups, we reveal hidden structures in the sets of weight multiplicities and obtain explicit closed formulas for the weight multiplicities. In particular we show that some special families of weight multiplicities form the Pascal, Catalan, Motzkin, Riordan and Bessel triangles.- Susumu Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), no. 4, 789–808. MR 1443748, DOI 10.1215/kjm/1250518452
- Georgia Benkart and Tom Halverson, Motzkin algebras, European J. Combin. 36 (2014), 473–502. MR 3131911, DOI 10.1016/j.ejc.2013.09.010
- Persi Diaconis and Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994), 49–62. Studies in applied probability. MR 1274717, DOI 10.2307/3214948
- Sen-Peng Eu, Skew-standard tableaux with three rows, Adv. in Appl. Math. 45 (2010), no. 4, 463–469. MR 2679926, DOI 10.1016/j.aam.2010.03.004
- Sen-Peng Eu, Tung-Shan Fu, Justin T. Hou, and Te-Wei Hsu, Standard Young tableaux and colored Motzkin paths, J. Combin. Theory Ser. A 120 (2013), no. 7, 1786–1803. MR 3092698, DOI 10.1016/j.jcta.2013.06.007
- Peter J. Forrester and S. Ole Warnaar, The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 4, 489–534. MR 2434345, DOI 10.1090/S0273-0979-08-01221-4
- Igor B. Frenkel and Mikhail G. Khovanov, Canonical bases in tensor products and graphical calculus for $U_q({\mathfrak {s}}{\mathfrak {l}}_2)$, Duke Math. J. 87 (1997), no. 3, 409–480. MR 1446615, DOI 10.1215/S0012-7094-97-08715-9
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257–285. MR 1041448, DOI 10.1016/0097-3165(90)90060-A
- Dominique Gouyou-Beauchamps, Standard Young tableaux of height $4$ and $5$, European J. Combin. 10 (1989), no. 1, 69–82. MR 977181, DOI 10.1016/S0195-6698(89)80034-4
- Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971, DOI 10.1090/gsm/042
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR 499562
- Rebecca L. Jayne and Kailash C. Misra, On multiplicities of maximal weights of $\widehat {sl}(n)$-modules, Algebr. Represent. Theory 17 (2014), no. 4, 1303–1321. MR 3228490, DOI 10.1007/s10468-014-9470-2
- Rebecca L. Jayne and Kailash C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb. 22 (2018), no. 1, 147–156. MR 3767671, DOI 10.1007/s00026-018-0374-4
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- Victor G. Kac and Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53 (1984), no. 2, 125–264. MR 750341, DOI 10.1016/0001-8708(84)90032-X
- Joel Kamnitzer, The crystal structure on the set of Mirković-Vilonen polytopes, Adv. Math. 215 (2007), no. 1, 66–93. MR 2354986, DOI 10.1016/j.aim.2007.03.012
- Seok-Jin Kang, Crystal bases for quantum affine algebras and combinatorics of Young walls, Proc. London Math. Soc. (3) 86 (2003), no. 1, 29–69. MR 1971463, DOI 10.1112/S0024611502013734
- Seok-Jin Kang and Jae-Hoon Kwon, Fock space representations of quantum affine algebras and generalized Lascoux-Leclerc-Thibon algorithm, J. Korean Math. Soc. 45 (2008), no. 4, 1135–1202. MR 2422732, DOI 10.4134/JKMS.2008.45.4.1135
- Masaki Kashiwara, Crystalizing the $q$-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249–260. MR 1090425
- Nan Hua Xi, Special bases of irreducible modules of the quantized universal enveloping algebra $U_v(\textrm {gl}(n))$, J. Algebra 154 (1993), no. 2, 377–386. MR 1206127, DOI 10.1006/jabr.1993.1020
- Masaki Kashiwara and Toshiki Nakashima, Crystal graphs for representations of the $q$-analogue of classical Lie algebras, J. Algebra 165 (1994), no. 2, 295–345. MR 1273277, DOI 10.1006/jabr.1994.1114
- S. Kass, R. V. Moody, J. Patera, and R. Slansky, Affine Lie algebras, weight multiplicities, and branching rules. Vols. 1, 2, Los Alamos Series in Basic and Applied Sciences, vol. 9, University of California Press, Berkeley, CA, 1990. MR 1117679
- Jang Soo Kim and Suho Oh, The Selberg integral and Young books, J. Combin. Theory Ser. A 145 (2017), 1–24. MR 3551643, DOI 10.1016/j.jcta.2016.07.005
- Jang Soo Kim and Dennis Stanton, On $q$-integrals over order polytopes, Adv. Math. 308 (2017), 1269–1317. MR 3600087, DOI 10.1016/j.aim.2017.01.001
- Young-Hun Kim, Se-jin Oh, and Young-Tak Oh, Cyclic sieving phenomenon on dominant maximal weights over affine Kac-Moody algebras, Adv. Math. 374 (2020), 107336, 75. MR 4133517, DOI 10.1016/j.aim.2020.107336
- Kazuhiko Koike, On new multiplicity formulas of weights of representations for the classical groups, J. Algebra 107 (1987), no. 2, 512–533. MR 885808, DOI 10.1016/0021-8693(87)90100-1
- Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205–263. MR 1410572
- Cédric Lecouvey and Mark Shimozono, Lusztig’s $q$-analogue of weight multiplicity and one-dimensional sums for affine root systems, Adv. Math. 208 (2007), no. 1, 438–466. MR 2304324, DOI 10.1016/j.aim.2006.03.001
- J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition identities, Adv. in Math. 29 (1978), no. 1, 15–59. MR 501091, DOI 10.1016/0001-8708(78)90004-X
- Peter Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499–525. MR 1356780, DOI 10.2307/2118553
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- Se-jin Oh, The Andrews-Olsson identity and Bessenrodt insertion algorithm on Young walls, European J. Combin. 43 (2015), 8–31. MR 3266281, DOI 10.1016/j.ejc.2014.07.001
- Amitai Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math. 41 (1981), no. 2, 115–136. MR 625890, DOI 10.1016/0001-8708(81)90012-8
- Bruce E. Sagan, The symmetric group, 2nd ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001. Representations, combinatorial algorithms, and symmetric functions. MR 1824028, DOI 10.1007/978-1-4757-6804-6
- N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org.
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
- Shunsuke Tsuchioka, Catalan numbers and level 2 weight structures of $A_{p-1}^{(1)}$, New trends in combinatorial representation theory, RIMS Kôkyûroku Bessatsu, B11, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, pp. 145–154. MR 2562491
- Shunsuke Tsuchioka and Masaki Watanabe, Pattern avoidance seen in multiplicities of maximal weights of affine Lie algebra representations, Proc. Amer. Math. Soc. 146 (2018), no. 1, 15–28. MR 3723117, DOI 10.1090/proc/13597
- Hermann Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, NJ, 1939. MR 255