Let $λ= \sum_{i \in I_{0}} m_{i} \varpi_{i}$, 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 $\mathbb{B}(λ)$ be the crystal of all Lakshmibai-Seshadri paths of shape $λ$.
First, we give an explicit description of the decomposition of the crystal $\mathbb{B}(λ)$ 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 $\mathbb{B}_{0}(λ)$ of $\mathbb{B}(λ)$ containing the straight line $π_λ$ as a specified subcrystal of the affinization $\hat{\mathbb{B}(λ)_{\mathrm{cl}}}$ (with weight lattice $P$) of the crystal $\mathbb{B}(λ)_{\mathrm{cl}} \simeq \bigotimes_{i \in I_{0}} \bigl(\mathbb{B}(\varpi_{i})_{\mathrm{cl}} \bigr)^{\otimes m_{i}}$ (with weight lattice $P_{\mathrm{cl}} = P/(\mathbb{Q}δ\cap P)$, where $δ$ is the null root of $\mathfrak{g}$), which was studied in a previous paper.
Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and set $λ: = Λ_1 - Λ_2$, where $Λ_1, Λ_2$ are the fundamental weights for $\mathfrak{g}$; note that $λ$ is neither dominant nor antidominant. Let $\mathbb{B}(λ)$ be the crystal of all Lakshmibai-Seshadri paths of shape $λ$. We prove that (the crystal graph of) $\mathbb{B}(λ)$ is connected. Furthermore, we give an explicit description of Lakshmibai-Seshadri paths of shape $λ$.
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 $λ$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $\mathrm{QLS}(λ)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $λ$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w λ}(q, t)$ are explicitly described in terms of QLS paths of shape $λ$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $λ$, $μ$, we have an isomorphism $Θ: \mathrm{QLS}(λ+ μ) \rightarrow \mathrm{QLS}(λ) \otimes \mathrm{QLS}(μ)$ of crystal…
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.
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 posi…
In this paper, we give a combinatorial realization of the crystal basis of a quantum Weyl module over a quantum affine algebra of type $A_{2n}^{(2)}$, and a representation-theoretic interpretation of the specialization $P_λ^{A_{2n}^{(2)}} (q,0)$ of the symmetric Macdonald polynomial $P_λ^{A_{2n}^{(2)}} (q,t)$ at $t=0$, where $λ$ is a dominant weight and $P_λ^{A_{2n}^{(2)}}(q,t)$ denotes the specific specialization of the symmetric Macdonald-Koornwinder polynomial $P_λ(q,t_1, t_2, t_3, t_4, t_5)$. More precisely, as some results for untwisted affine types, the set of all ($A_{2n}^{(2)}$-type) q…
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.
Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and let $λ$ be an arbitrary integral weight. We denote by $\mathbb{B}(λ)$ the crystal of all Lakshmibai-Seshadri paths of shape $λ$. Let $V(λ)$ be the extremal weight module of extremal weight $λ$ generated by the (cyclic) extremal weight vector $v_λ$ of weight $λ$, and let $\mathcal{B}(λ)$ be the crystal basis of $V(λ)$ with $u_λ\in \mathcal{B}(λ)$ the element corresponding to $v_λ$. We prove that the connected component $\mathcal{B}_0(λ)$ of $\mathcal{B}(λ)$ containing $u_λ$ is isomorphic, as a crystal, to the connected compon…
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) qua…
In this paper, we give a characterization of the crystal bases $\mathcal{B}_{x}^{+}(λ)$, $x \in W_{\mathrm{af}}$, of Demazure submodules $V_{x}^{+}(λ)$, $x \in W_{\mathrm{af}}$, of a level-zero extremal weight module $V(λ)$ over a quantum affine algebra $U_{q}$, where $λ$ 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 $\mathcal{B}(λ)$ of the le…
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-Richardso…
We apply the crystal basis theory for Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type $G(p,p,n)$. This yields a classification of simple modules over these cyclotomic Hecke algebras in the non-separated case, generalizing our previous work [J. Hu, J. Algebra 267 (2003) 7-20]. The separated case was completed in [J. Hu, J. Algebra 274 (2004) 446--490]. Furthermore, we use Naito--Sagaki's work [S. Naito & D. Sagaki, J. Algebra 251 (2002) 461--474] on Lakshmibai--Seshadri paths fixed by diagram automorphisms to derive explicit formu…
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}…
In this paper, we give a representation-theoretic interpretation of the specialization $E_{w_{\circ} λ} (q,\infty)$ of the nonsymmetric Macdonald polynomial $E_{w_{\circ} λ}(q,t)$ at $t=\infty$ in terms of the Demazure submodule $V_{w_\circ}^{-} (λ)$ of the level-zero extremal weight module $V(λ)$ over a quantum affine algebra of an arbitrary untwisted type, here, $λ$ 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 $E_{x λ} (q, \infty)$ at $t=\infty$ of th…
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 …
Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set $λ=Λ_{1} - Λ_{2}$, where $Λ_{1}$, $Λ_{2}$ are the fundamental weights. Denote by $V(λ)$ the extremal weight module of extremal weight $λ$ with $v_λ$ the extremal weight vector, and by $\mathcal{B}(λ)$ the crystal basis of $V(λ)$ with $u_λ$ the element corresponding to $v_λ$. We prove that (i) $\mathcal{B}(λ)$ is connected, (ii) the subset $\mathcal{B}(λ)_μ$ of elements of weight $μ$ in $\mathcal{B}(λ)$ is a finite set for every integral weight $μ$, and $\mathcal{B}(λ)_λ = \{u_λ\}$, (iii) every extremal element in $\mathcal…
We establish the equality of the specialization $P_λ(x;q,0)$ of the Macdonald polynomial at $t=0$ with the graded character $X_λ(x;q)$ 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…
We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application…
