Bibliography

[Bourbaki46]

Nicolas Bourbaki. Lie Groups and Lie Algebras: Chapters 4-6. Springer, reprint edition, 1998.

[BumpNakasuji2010]

D. Bump and M. Nakasuji. Casselman’s basis of Iwahori vectors and the Bruhat order. arXiv 1002.2996, arXiv 1002.2996.

[BumpSchilling2017]

D. Bump and A. Schilling, Crystal bases: representations and combinatorics, World Scientific, 2017.

[Carrell1994]

J. B. Carrell. The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties. In Algebraic Groups and Their Generalizations: Classical Methods, AMS Proceedings of Symposia in Pure Mathematics, 56, 53–61, 1994.

[Deodhar1977]

V. V. Deodhar. Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Moebius function. Inventiones Mathematicae, 39(2):187–198, 1977.

[Dyer1993]

M. J. Dyer. The nil Hecke ring and Deodhar’s conjecture on Bruhat intervals. Inventiones Mathematicae, 111(1):571–574, 1993.

[Dynkin1952]

E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras. (Russian) Mat. Sbornik N.S. 30(72):349–462, 1952.

[FauserEtAl2006]

B. Fauser, P. D. Jarvis, R. C. King, and B. G. Wybourne. New branching rules induced by plethysm. Journal of Physics A. 39(11):2611–2655, 2006.

[Fulton1997]

W. Fulton. Young Tableaux. Cambridge University Press, 1997.

[FourierEtAl2009]

G. Fourier, M. Okado, A. Schilling. Kirillov–Reshetikhin crystal for nonexceptional types. Advances in Mathematics, 222:1080–1116, 2009.

[FourierEtAl2010]

G. Fourier, M. Okado, A. Schilling. Perfectness of Kirillov-Reshetikhin crystals for nonexceptional types. Contemp. Math., 506:127–143, 2010.

[HatayamaEtAl2001]

G. Hatayama, A. Kuniba, M. Okado, T. Takagi, Z. Tsuboi. Paths, crystals and fermionic formulae. in MathPhys Odyssey 2001, in : Prog. Math. Phys., vol 23, Birkhauser Boston, Boston, MA 2002, pp. 205–272.

[HainesEtAl2009]

T. J. Haines, R. E. Kottwitz, and A. Prasad. Iwahori-Hecke Algebras. arXiv math/0309168.

[HongKang2002]

J. Hong and S.-J. Kang. Introduction to Quantum Groups and Crystal Bases. AMS Graduate Studies in Mathematics, American Mathematical Society, 2002.

[HongLee2008]

J. Hong and H. Lee. Young tableaux and crystal \(B(\infty)\) for finite simple Lie algebras. J. Algebra, 320:3680–3693, 2008.

[HoweEtAl2005]

R. Howe, E.-C.Tan, and J. F. Willenbring. Stable branching rules for classical symmetric pairs. Transactions of the American Mathematical Society, 357(4):1601–1626, 2005.

[Iwahori1964]

N. Iwahori. On the structure of a Hecke ring of a Chevalley group over a finite field. J. Fac. Sci. Univ. Tokyo Sect. I, 10:215–236, 1964.

[JayneMisra2014]

R. Jayne and K. Misra, On multiplicities of maximal weights of \(\widehat{sl}(n)\)-modules. Algebr. Represent. Theory 17 (2014), no. 4, 1303–1321. arXiv 1309.4969.

[Jimbo1986]

M. A. Jimbo. \(q\)-analogue of \(U(\mathfrak{gl}(N+1))\), Hecke algebra, and the Yang-Baxter equation. Lett. Math. Phys, 11(3):247–252, 1986.

[JonesEtAl2010]

B. Jones, A. Schilling. Affine structures and a tableau model for E_6 crystals J. Algebra, 324:2512-2542, 2010.

[Joseph1995]

A. Joseph. Quantum Groups and Their Primitive Ideals. Springer-Verlag, 1995.

[Kac]

Victor G. Kac. Infinite Dimensional Lie algebras, Cambridge University Press, third edition, 1994.

[KacPeterson]

Kac and Peterson. Infinite-dimensional Lie algebras, theta functions and modular forms. Adv. in Math. 53 (1984), no. 2, 125-264.

[KKMMNN1992]

S-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, A. Nakayashiki. Affine crystals and vertex models. Int. J. Mod. Phys. A 7 (suppl. 1A): 449–484, 1992.

[KKS2007]

S.-J. Kang, J.-A. Kim, and D.-U. Shin. Modified Nakajima monomials and the crystal \(B(\infty)\). J. Algebra, 308 (2007), 524-535.

[Kashiwara1993]

M. Kashiwara. The crystal base and Littelmann’s refined Demazure character formula. Duke Math. J., 71(3):839–858, 1993.

[Kashiwara1995]

M. Kashiwara. On crystal bases. Representations of groups (Banff, AB, 1994), 155–197, CMS Conference Proceedings, 16, American Mathematical Society, Providence, RI, 1995.

[KashiwaraNakashima1994]

M. Kashiwara and T. Nakashima. Crystal graphs for representations of the \(q\)-analogue of classical Lie algebras. Journal Algebra, 165(2):295–345, 1994.

[KMPS]

Kass, Moody, Patera and Slansky, Affine Lie algebras, weight multiplicities, and branching rules. Vols. 1, 2. University of California Press, Berkeley, CA, 1990.

[KimShin2010]

J.-A. Kim and D.-U. Shin. Generalized Young walls and crystal bases for quantum affine algebra of type \(A\). Proc. Amer. Math. Soc., 138(11):3877–3889, 2010.

[KimLeeOh2017]

Jang Soo Kim, Kyu-Hwan Lee and Se-Jin Oh, Weight multiplicities and Young tableaux through affine crystals. arXiv 1703.10321 (2017).

[King1975]

R. C. King. Branching rules for classical Lie groups using tensor and spinor methods. Journal of Physics A, 8:429–449, 1975.

[Knuth1970]

D. Knuth. Permutations, matrices, and generalized Young tableaux. Pacific Journal of Mathematics, 34(3):709–727, 1970.

[Knuth1998]

D. Knuth. The Art of Computer Programming. Volume 3. Sorting and Searching. Addison Wesley Longman, 1998.

[LNSSS14I]

C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono. A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph. (2014) arXiv 1211.2042

[LNSSS14II]

C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono. A uniform model for Kirillov-Reshetikhin crystals II: Alcove model, path model, and \(P = X\). (2014) arXiv 1402.2203

[L1995]

P. Littelmann. Paths and root operators in representation theory. Ann. of Math. (2) 142 (1995), no. 3, 499-525.

[Macdonald2003]

I. Macdonald. Affine Hecke algebras and orthogonal polynomials, Cambridge, 2003.

[McKayPatera1981]

W. G. McKay and J. Patera. Tables of Dimensions, Indices and Branching Rules for Representations of Simple Lie Algebras. Marcel Dekker, 1981.

[OkadoSchilling2008]

M. Okado, A.Schilling. Existence of crystal bases for Kirillov–Reshetikhin crystals for nonexceptional types. Representation Theory 12:186–207, 2008.

[Seitz1991]

G. Seitz, Maximal subgroups of exceptional algebraic groups. Mem. Amer. Math. Soc. 90 (1991), no. 441.

[Rubenthaler2008]

H. Rubenthaler, The (A2,G2) duality in E6, octonions and the triality principle. Trans. Amer. Math. Soc. 360 (2008), no. 1, 347–367.

[SalisburyScrimshaw2015]

B. Salisbury and T. Scrimshaw. A rigged configuration model for \(B(\infty)\). J. Combin. Theory Ser. A, 133:29–57, 2015.

[Schilling2006]

A. Schilling. Crystal structure on rigged configurations. Int. Math. Res. Not., Volume 2006. (2006) Article ID 97376. Pages 1-27.

[SchillingTingley2011]

A. Schilling, P. Tingley. Demazure crystals, Kirillov-Reshetikhin crystals, and the energy function. preprint arXiv 1104.2359

[Stanley1999]

R. P. Stanley. Enumerative Combinatorics, Volume 2. Cambridge University Press, 1999.

[Testerman1989]

Testerman, Donna M. A construction of certain maximal subgroups of the algebraic groups E6 and F4. J. Algebra 122 (1989), no. 2, 299–322.

[Testerman1992]

Testerman, Donna M. The construction of the maximal A1’s in the exceptional algebraic groups. Proc. Amer. Math. Soc. 116 (1992), no. 3, 635–644.