============ 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 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 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.