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In our previous paper, we gave a presentation of the torus-equivariant quantum $K$-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum $K$-theory ring $QK_{H}(Fl_{n+1})$ under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant $K$-group of the semi-infinite flag manifold associated to $SL_{n+1}(\mathbb{C})$., Comment: arXiv admin note: text overlap with arXiv:2302.09485

We give a presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, as a quotient of a polynomial ring by an explicit ideal. This is the torus-equivariant version of our previous result, which gives a presentation of the non-equivariant quantum $K$-theory ring of flag manifolds of type $A$. However, the method of proof for the torus-equivariant one is completely different from that for the non-equivariant one; our proof is based on the result in the $Q = 0$ limit, and uses Nakayama-type arguments to upgrade it to the quantum situation. Also, in contrast to the non-equivariant case in which we used the Chevalley formula, we make use of the inverse Chevalley formula for the torus-equivariant $K$-group of semi-infinite flag manifolds to obtain a relation which yields our presentation.

The purpose of this paper is to prove a Pieri-type multiplication formula for quantum Grothendieck polynomials, which was conjectured by Lenart-Maeno. This formula would enable us to compute explicitly the quantum product of two arbitrary (opposite) Schubert classes in the (small) quantum $K$-theory ring $QK(Fl_{n})$ of the (full) flag manifold $Fl_{n}$ of type $A_{n-1}$ on the basis of the fact that quantum Grothendieck polynomials represent (opposite) Schubert classes in $QK(Fl_{n})$.

In this paper we derive a counterpart of the well-known Ram-Yip formula for symmetric and nonsymmetric Macdonald polynomials of arbitrary type. Our new formula is in terms of a generalization of the Lakshmibai-Seshadri paths (originating in standard monomial theory), which we call pseudo-quantum Lakshmibai-Seshadri (LS) paths. This model carries less information than the alcove walks in the Ram-Yip formula, and it is therefore more efficient. Furthermore, we construct a connected pseudo-crystal structure on the pseudo-quantum LS paths, which is expected to lead to a simple Littlewood-Richardson rule for multiplying Macdonald polynomials. By contrast with the Kashiwara crystals, our pseudo-crystals have edges labeled by arbitrary roots.

We continue the study, begun in [Kouno-Naito-Orr-Sagaki, 2021], 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 for $E_8$).

We derive cancellation-free Chevalley-type multiplication formulas in the T-equivariant quantum K-theory 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 of arbitrary flag manifolds G/B, which was derived earlier in terms of the quantum alcove model, by the last three authors., Comment: We added the proof of the positivity property for the Chevalley structure constants for full flag manifolds of arbitrary types, two-step flag manifolds of type A, and Grassmannians of type C. We also added Appendix A (a new proof of the existence of a multiplicative surjection due to Kato) and Appendix B (on the connection with a conjecture of Weihong Xu, who is a co-author of this Appendix)

We give a Chevalley formula for an arbitrary weight for the torus-equivariant $K$-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum $K$-theory $QK_{T}(G/B)$ of an (ordinary) flag manifold $G/B$; this has been a longstanding conjecture about the multiplicative structure of $QK_{T}(G/B)$. In type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum $K$-theory $QK(SL_{n}/B)$; we also obtain very explicit information about the coefficients in the respective Chevalley formula., Comment: arXiv admin note: text overlap with arXiv:1911.12773

We prove an explicit inverse Chevalley formula in the equivariant $K$-theory of semi-infinite flag manifolds of simply-laced type. By an inverse Chevalley formula, we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb{Z}[q^{\pm 1}]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply-laced type and equivariant scalars $e^{\lambda}$, where $\lambda$ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply-laced type, except for type $E_8$. The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. As such, our formula also provides an explicit determination of all nonsymmetric $q$-Toda operators for minuscule weights in ADE type.

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 the anti-dominant minuscule fundamental weight $- \varpi_{k}$ in the torus-equivariant quantum $K$-group of the partial flag manifold $G/P_{J}$ (where $J = I \setminus \{k\}$) 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 $\mathbf{Q}_{J}$ 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 $\mathbf{Q}_{J}$, recently established by Kato., Comment: 30 pages

We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-theory of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum K-theory of the flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of the mentioned quantum K-theory. Moreover, in type A, we prove that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivariant) quantum K-theory of the corresponding flag manifold.

We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or, equivalently, quantum Lakshmibai-Seshadri paths). In view of recent results of Kato, these formulas give an explicit description of the structure constants for the Pontryagin product in the torus-equivariant $K$-group of affine Grassmannians and that for the quantum multiplication of the torus-equivariant (small) quantum $K$-group of finite-dimensional flag manifolds. Our proof of these formulas is based on standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, which is established in our previous work, and also uses a string property of Demazure-like subsets of the crystal basis of a level-zero extremal weight module over a quantum affine algebra.

Let $\lambda$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $\mathrm{QLS}(\lambda)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $\lambda$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w \lambda}(q, t)$ are explicitly described in terms of QLS paths of shape $\lambda$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $\lambda$, $\mu$, we have an isomorphism $\Theta : \mathrm{QLS}(\lambda + \mu) \rightarrow \mathrm{QLS}(\lambda) \otimes \mathrm{QLS}(\mu)$ of crystals. In this paper, we study the behavior of the degree function under the isomorphism $\Theta$ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules., Comment: arXiv admin note: text overlap with arXiv:1802.06339

Let $\lambda \in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W/W_{\lambda}$, where $W_{\lambda}$ is the stabilizer of $\lambda$ in a finite Weyl group $W$. In this paper, we give a module $\mathbb{K}_{w}(\lambda)$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w \lambda}(q,\,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1 - q^{-r})^{-1}$ for a positive integer $r$. This module $\mathbb{K}_{w}(\lambda)$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $V_{w}^{-}(\lambda)$ by the sum of the submodules $V_{z}^{-}(\lambda)$ for all those coset representatives $z$ of minimal length for the cosets in $W/W_{\lambda}$ such that $z > w$ in the Bruhat order $<$ on $W$.

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set $\lambda=\Lambda_{1} - \Lambda_{2}$, where $\Lambda_{1}$, $\Lambda_{2}$ are the fundamental weights. Denote by $V(\lambda)$ the extremal weight module of extremal weight $\lambda$ with $v_\lambda$ the extremal weight vector, and by $\mathcal{B}(\lambda)$ the crystal basis of $V(\lambda)$ with $u_\lambda$ the element corresponding to $v_\lambda$. We prove that (i) $\mathcal{B}(\lambda)$ is connected, (ii) the subset $\mathcal{B}(\lambda)_{\mu}$ of elements of weight $\mu$ in $\mathcal{B}(\lambda)$ is a finite set for every integral weight $\mu$, and $\mathcal{B}(\lambda)_{\lambda} = \{u_\lambda\}$, (iii) every extremal element in $\mathcal{B}(\lambda)$ is contained in the Weyl group orbit of $u_\lambda$, (iv) $V(\lambda)$ is irreducible. Finally, we prove that the crystal basis $\mathcal{B}(\lambda)$ is isomorphic, as a crystal, to the crystal $\mathbb{B}(\lambda)$ of Lakshmibai-Seshadri paths of shape $\lambda$., Comment: 20 pages, 2 diagrams

We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $\mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in the $K$-theory of $\mathbf{Q}_{G}$, of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over $\mathbf{Q}_{G}$. In order to achieve this, we provide a number of fundamental results on $\mathbf{Q}_{G}$ and its Schubert subvarieties including the Borel-Weil-Bott theory, whose special case is conjectured in [A. Braverman and M. Finkelberg, Weyl modules and $q$-Whittaker functions, Math. Ann. 359 (2014), 45--59]. One more ingredient of this paper besides the geometric results above is (a combinatorial version of) standard monomial theory for level-zero extremal weight modules over quantum affine algebras, which is described in terms of semi-infinite Lakshmibai-Seshadri paths. In fact, in our Pieri-Chevalley formula, the positivity of structure coefficients is proved by giving an explicit representation-theoretic meaning through semi-infinite Lakshmibai-Seshadri paths.

In this paper, we give a representation-theoretic interpretation of the specialization $E_{w_{\circ} \lambda} (q,\infty)$ of the nonsymmetric Macdonald polynomial $E_{w_{\circ} \lambda}(q,t)$ at $t=\infty$ in terms of the Demazure submodule $V_{w_\circ}^{-} (\lambda)$ of the level-zero extremal weight module $V(\lambda)$ over a quantum affine algebra of an arbitrary untwisted type, here, $\lambda$ is a dominant integral weight, and $w_{\circ}$ denotes the longest element in the finite Weyl group $W$. Also, for each $x \in W$, we obtain a combinatorial formula for the specialization $E_{x \lambda} (q, \infty)$ at $t=\infty$ of the nonsymmetric Macdonald polynomial $E_{x \lambda} (q,t)$, and also one for the graded character $\mathrm{gch} V_{x}^- (\lambda)$ of the Demazure submodule $V_{x}^- (\lambda)$ of $V(\lambda)$, both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape $\lambda$., Comment: 45 pages

We establish the equality of the specialization $E_{w\lambda}(x;q,0)$ of the nonsymmetric Macdonald polynomial $E_{w\lambda}(x;q,t)$ at $t=0$ with the graded character $\mathop{\rm gch} U_{w}^{+}(\lambda)$ of a certain Demazure-type submodule $U_{w}^{+}(\lambda)$ of a tensor product of "single-column" Kirillov--Reshetikhin modules for an untwisted affine Lie algebra, where $\lambda$ is a dominant integral weight and $w$ is a (finite) Weyl group element, this generalizes our previous result, that is, the equality between the specialization $P_{\lambda}(x;q,0)$ of the symmetric Macdonald polynomial $P_{\lambda}(x;q,t)$ at $t=0$ and the graded character of a tensor product of single-column Kirillov--Reshetikhin modules. We also give two combinatorial formulas for the mentioned specialization of a nonsymmetric Macdonald polynomial: one in terms of quantum Lakshmibai-Seshadri paths and the other in terms of the quantum alcove model., Comment: updated references

We give an explicit and computable description, in terms of the parabolic quantum Bruhat graph, of the degree function defined for quantum Lakshmibai-Seshadri paths, or equivalently, for "projected" (affine) level-zero Lakshmibai-Seshadri paths. This, in turn, gives an explicit and computable description of the global energy function on tensor products of Kirillov-Reshetikhin crystals of one-column type, and also of (classically restricted) one-dimensional sums., Comment: arXiv admin note: text overlap with arXiv:1308.3529

In this paper, we give a characterization of the crystal bases $\mathcal{B}_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of Demazure submodules $V_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of a level-zero extremal weight module $V(\lambda)$ over a quantum affine algebra $U_{q}$, where $\lambda$ is an arbitrary level-zero dominant integral weight, and $W_{\mathrm{af}}$ denotes the affine Weyl group. This characterization is given in terms of the initial direction of a semi-infinite Lakshmibai-Seshadri path, and is established under a suitably normalized isomorphism between the crystal basis $\mathcal{B}(\lambda)$ of the level-zero extremal weight module $V(\lambda)$ and the crystal $\mathbb{B}^{\frac{\infty}{2}}(\lambda)$ of semi-infinite Lakshmibai-Seshadri paths of shape $\lambda$, which is obtained in our previous work. As an application, we obtain a formula expressing the graded character of the Demazure submodule $V_{w_0}^{+}(\lambda)$ in terms of the specialization at $t=0$ of the symmetric Macdonald polynomial $P_{\lambda}(x\,;\,q,\,t)$.

We introduce semi-infinite Lakshmibai-Seshadri paths by using the semi-infinite Bruhat order (or equivalently, Lusztig's generic Bruhat order) on affine Weyl groups in place of the usual Bruhat order. These paths enable us to give an explicit realization of the crystal basis of an extremal weight module of an arbitrary level-zero dominant integral extremal weight over a quantum affine algebra. This result can be thought of as a full generalization of our previous result (which uses Littelmann's Lakshmibai-Seshadri paths), in which the level-zero dominant integral weight is assumed to be a positive-integer multiple of a level-zero fundamental weight.

We establish the equality of the specialization $P_\lambda(x;q,0)$ of the Macdonald polynomial at $t=0$ with the graded character $X_\lambda(x;q)$ of a tensor product of "single-column" Kirillov-Reshetikhin (KR) modules for untwisted affine Lie algebras. This is achieved by constructing two uniform combinatorial models for the crystals associated with the mentioned tensor products: the quantum alcove model (which is naturally associated to Macdonald polynomials), and the quantum Lakshmibai-Seshadri path model. We provide an explicit affine crystal isomorphism between the two models, and realize the energy function in both models. In particular, this gives the first proof of the positivity of the $t = 0$ limit of the symmetric Macdonald polynomial in the untwisted and non-simply-laced cases, when it is expressed as a linear combination of the irreducible characters for a finite-dimensional simple Lie subalgebra, as well as a representation-theoretic meaning of the coefficients in this expression in terms of degree functions., Comment: 43 pages, 1 figure; slight reorganization of the material, additional details added

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 [M] and [SS] 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., Comment: to appear in Mathematische Zeitschrift

We give an explicit description of the image of a quantum LS path, regarded as a rational path, under the action of root operators, and show that the set of quantum LS paths is stable under the action of the root operators. As a by-product, we obtain a new proof of the fact that a projected level-zero LS path is just a quantum LS path.

By applying Miyamoto's $\mathbb{Z}_{3}$-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices and their automorphisms of order 3, we construct holomorphic vertex operator algebras of central charge 24 whose Lie algebras of the weight one spaces are of types $A_{2,3}^6$, $E_{6,3}G_{2,1}^{3}$, and $A_{5,3}D_{4,3}A_{1,1}^{3}$, which correspond to No.6, No.17, and No.32 on Schellekens' list, respectively.

We present a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai-Seshadri paths (in the theory of the Littelmann path model). This generalization is based on the graph on parabolic cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related model is the so-called quantum alcove model. The proof is based on two lifts of the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl group and to Littelmann's poset on level-zero weights. Our construction leads to a simple calculation of the energy function. It also implies the equality between a Macdonald polynomial specialized at t=0 and the graded character of a tensor product of KR modules., Comment: 10 pages, 1 figure

We lift the parabolic quantum Bruhat graph into the Bruhat order on the affine Weyl group and into Littelmann's poset on level-zero weights. We establish a quantum analogue of Deodhar's Bruhat-minimum lift from a parabolic quotient of the Weyl group. This result asserts a remarkable compatibility of the quantum Bruhat graph on the Weyl group, with the cosets for every parabolic subgroup. Also, we generalize Postnikov's lemma from the quantum Bruhat graph 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., Comment: 36 pages; 3 figures; Section 8 added; version to appear in IMRN

Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight \lambda, we define a preorder on the set $P(\lambda, k)$ of k-tuples of dominant weights which add up to \lambda. Let $P(\lambda, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if \lambda is a multiple of a fundamental weight (and k is general) or if k=2 (and \lambda is general), then $P(\lambda, k)/\sim$ coincides with the set of S_k-orbits in $P(\lambda,k)$, where S_k acts on $P(\lambda, k)$ as the permutations of components. If g is of type A_n and k=2, we show that the S_2-orbit of the row shuffle defined by Fomin et al is the unique maximal element in the poset. Given an element of $P(\lambda, k)$, consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, \lambda, and k) the dimension of this tensor product increases along with the partial order. We also show that in the case when \lambda is a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A_2 and k=2 (\lambda is general), there exists an inclusion of tensor products of g-modules along with the partial order. In particular, if g is of type A_n, this means that the difference of the characters is Schur positive., Comment: 24 pages, updated references

We prove the connectedness of the crystal, which we introduced in our previous works., Comment: 32 pages, no figure

In the present paper, we give an explicit description of the affine analogs of Berenstein-Zelevinsky data constructed in our previous paper: Toward Berenstein-Zelevinsky data in affine type $A$, I: Construction of affine analogs (arXiv:1009.4526), in terms of certain collections of nonnegative integers, which we call Lusztig data of type $A_{l-1}^{(1)}$., Comment: 29 pages

The purpose of this paper is to prove that the Mirkovic-Vilonen (MV for short) polytope corresponding to the tensor product of two arbitrary MV polytopes is contained in the Minkowski sum of these two MV polytopes. This generalizes the result in our previous paper [KNS], which was obtained under the assumption that the first tensor factor is an extremal MV polytope., Comment: 20 pages

We give (conjectural) analogs of Berenstein-Zelevinsky data for affine type $A$. Moreover, by using these affine analogs of Berenstein-Zelevinsky data, we realize the crystal basis of the negative part of the quantized universal enveloping algebra of the (Langlands dual) Lie algebra of affine type $A$., Comment: 46 pages

Using Lakshmibai-Seshadri paths, we give a combinatorial realization of the crystal basis of an extremal weight module of integral extremal weight over the quantized universal enveloping algebra associated to the infinite rank affine Lie algebra of type $B_{\infty}$, $C_{\infty}$, or $D_{\infty}$. 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 $B_{\infty}$, $C_{\infty}$, and $D_{\infty}$, extend the corresponding results due to Kwon, in types $A_{+\infty}$ and $A_{\infty}$; our results above also include, as a special case, the corresponding results (concerning crystal bases) due to Lecouvey, in types $B_{\infty}$, $C_{\infty}$, and $D_{\infty}$, where the extremal weights are of level zero.

In this paper, we give a polytopal estimate of Mirkovi\'c-Vilonen polytopes lying in a Demazure crystal in terms of Minkowski sums of extremal Mirkovi\'c-Vilonen polytopes. As an immediate consequence of this result, we provide a necessary (but not sufficient) polytopal condition for a Mirkovi\'c-Vilonen polytope to lie in a Demazure crystal.

Let $r \in \BC$ be a complex number, and $d \in \BZ_{\ge 2}$ a positive integer greater than or equal to 2. Ashihara and Miyamoto introduced a vertex operator algebra $\Vam$ of central charge $dr$, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size $d$. In this paper, we prove that the vertex operator algebra $\Vam$ is simple if and only if $r$ is not an integer. Further, in the case that $r$ is an integer (i.e., $\Vam$ is not simple), we give a generator system of the maximal proper ideal $I_{r}$ of the VOA $\Vam$ explicitly., Comment: 30 pages, no figure

We give a necessary and sufficient condition for an MV polytope $P$ in a highest weight crystal to lie in an arbitrary fixed Demazure crystal (resp., opposite Demazure crystal), in terms of the lengths of edges along a path through the 1-skeleton of $P$ corresponding to a reduced word for the longest element of the Weyl group $W$. % Also, we give an explicit description as a pseudo-Weyl polytope for extremal MV polytopes in a highest weight crystal. % Finally, by combining the results above, we obtain a polytopal condition for an MV polytope $P$ to lie in an arbitrary fixed opposite Demazure crystal.

We give an explicit description of the (lowering) Kashiwara operators on Mirkovi\'c-Vilonen polytopes in types $B$ and $C$, which provides a simple method for generating Mirkovi\'c-Vilonen polytopes inductively. This description can be thought of as a modification of the original Anderson-Mirkovi\'c conjecture, which Kamnitzer proved in the case of type $A$, and presented a counterexample in the case of type $C_{3}$.

We give interpretations of energy functions and (classically restricted) one-dimensional sums associated to tensor products of level-zero fundamental representations of quantum affine algebras in terms of Lakshmibai-Seshadri paths of level-zero weight shape.

Let $\lambda = \sum_{i \in I_{0}} m_{i} \varpi_{i}$, with $m_{i} \in \mathbb{Z}_{\ge 0}$ for $i \in I_{0}$, be a level-zero dominant integral weight for an affine Lie algebra $\mathfrak{g}$ over $\mathbb{Q}$, where the $\varpi_{i}$, $i \in I_{0}$, are the level-zero fundamental weights, and let $\mathbb{B}(\lambda)$ be the crystal of all Lakshmibai-Seshadri paths of shape $\lambda$. First, we give an explicit description of the decomposition of the crystal $\mathbb{B}(\lambda)$ into a disjoint union of connected components, and show that all the connected components are pairwise ``isomorphic'' (up to a shift of weights). Second, we ``realize'' the connected component $\mathbb{B}_{0}(\lambda)$ of $\mathbb{B}(\lambda)$ containing the straight line $\pi_{\lambda}$ as a specified subcrystal of the affinization $\hat{\mathbb{B}(\lambda)_{\mathrm{cl}}}$ (with weight lattice $P$) of the crystal $\mathbb{B}(\lambda)_{\mathrm{cl}} \simeq \bigotimes_{i \in I_{0}} \bigl(\mathbb{B}(\varpi_{i})_{\mathrm{cl}} \bigr)^{\otimes m_{i}}$ (with weight lattice $P_{\mathrm{cl}} = P/(\mathbb{Q}\delta \cap P)$, where $\delta$ is the null root of $\mathfrak{g}$), which was studied in a previous paper.

Assuming the existence of the perfect crystal bases of Kirillov-Reshetikhin modules over simply-laced quantum affine algebras, we construct certain perfect crystals for twisted quantum affine algebras, and also provide compelling evidence that the constructed crystals are isomorphic to the conjectural crystal bases of Kirillov-Reshetikhin modules over twisted quantum affine algebras.