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We classify, up to conjugation, all automorphisms of Niemeier lattices to which we can apply Miyamoto's orbifold construction. Using this classification, we prove that the VOAs obtained in Miyamoto (Symmetries, integrable systems and representations. Springer proceedings in mathematics and statistics, vol 40. Springer, London, pp 319-344, ) and Sagaki and Shimakura (Trans Am Math Soc, to appear) are all of holomorphic non-lattice VOAs which we can obtain by applying the $${\mathbb {Z}}_3$$ -orbifold construction to a Niemeier lattice and its automorphism. [ABSTRACT FROM AUTHOR]

We lift the parabolic quantum Bruhat graph (QBG) into the Bruhat order on the affine Weyl group and into Littelmann's poset on level-zero weights. We establish a quantum analog of Deodhar's Bruhat-minimum lift from a parabolic quotient of the Weyl group. This result asserts a remarkable compatibility of the QBG on the Weyl group, with the cosets for every parabolic subgroup. Also, we generalize Postnikov's lemma from the QBG to the parabolic one; this lemma compares paths between two vertices in the former graph. The results in this paper will be applied in a second paper to establish a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals, and the equality, for untwisted affine root systems, between the Macdonald polynomial with t set to zero and the graded character of tensor products of one-column KR modules. [ABSTRACT FROM AUTHOR]

Using Lakshmibai-Seshadri paths, we give a combinatorial realization of the crystal basis of an extremal weight module of a (general) integral extremal weight over the quantized universal enveloping algebra associated to the infinite rank affine Lie algebra of type B∞, C∞, or D∞. Moreover, via this realization, we obtain an explicit description (in terms of Littlewood- Richardson coefficients) of how tensor products of these crystal bases decompose into connected components when their extremal weights are of nonnegative levels. These results in types B∞, C∞, and D∞ extend the corresponding results due to Kwon in types A+∞ and A∞. Our results above also include, as a special case, the corresponding results (concerning crystal bases) due to Lecouvey in types B∞, C∞, and D∞, where the extremal weights are of level zero. [ABSTRACT FROM AUTHOR]

Let λ=∑i∈I0miϖi with mi ∈ ℤ≥0 be an integral weight of level zero, where the ϖi, i ∈ I0, are the level-zero fundamental weights for an affine Lie algebra g over ℚ, and let B(λ) be the crystal (with weight lattice P) of all Lakshmibai-Seshadri paths of shape λ. We define a crystal B(λ)cl (with weight lattice Pcl ≔ P/(ℚ δ∩ P)) to be the set of paths cl(π) : [0, 1] → ℚ ⊗ℤ Pcl for π ∈ B(λ), where the path cl(π) is defined to be the composition of the path π :[0,1] → ℚ ⊗ℤ P with the canonical projection cl:ℚ⊗ℤP↠ℚ⊗ℤPcl. We prove that this crystal B(\λ)cl is isomorphic to the tensor product ⊗i∈I0(B(ϖi)cl)⊗mi of the crystals B(ϖi)cl, i ∈ I0, as a crystal with weight lattice Pcl. As a corollary, we obtain that for each integral weight λ ∈ P of level zero, the crystal B(λ)cl is isomorphic, as a crystal with weight lattice Pcl, to the crystal base of a (certain) tensor product of the fundamental modules W(ϖi), i ∈ I0, of level zero over the quantized universal enveloping algebra Uq′(g) of g over ℚ(q) with weight lattice Pcl. [ABSTRACT FROM AUTHOR]

We give a path model for a level-zero extremal weight module over a quantum affine algebra. By using this result, we prove branching rules for an extremal weight module and a fundamental representation of level zero with respect to the quantized universal enveloping algebra of a Levi subalgebra. [ABSTRACT FROM AUTHOR]

We derive cancellation-free Chevalley-type multiplication formulas for the T -equivariant quantum K -theory ring of Grassmannians of type A and C , and also those of two-step flag manifolds of type A. They are obtained based on the uniform Chevalley formula in the T -equivariant quantum K -theory ring of arbitrary flag manifolds G / B , which was derived earlier in terms of the quantum alcove model, by the last three authors. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to an anti-dominant minuscule fundamental weight − ϖ k in the torus-equivariant quantum K -group of the partial flag manifold G / P J (where J = I ∖ { k }) corresponding to the maximal (standard) parabolic subgroup P J of minuscule type in type A , D , E , or B. This result is obtained by proving a similar formula in a torus-equivariant K -group of the semi-infinite partial flag manifold Q J of minuscule type, and then by making use of the isomorphism between the torus-equivariant quantum K -group of G / P J and the torus-equivariant K -group of Q J , recently established by Kato. [ABSTRACT FROM AUTHOR]

We continue the study, begun in [11] , of inverse Chevalley formulas for the equivariant K -group of semi-infinite flag manifolds. Using the language of alcove paths, we reformulate and extend our combinatorial inverse Chevalley formula to arbitrary weights in all simply-laced types (conjecturally also in type E 8). [ABSTRACT FROM AUTHOR]