Study on the fusion products and crystalbases of level-zero representations of quantum affine algebras

设g是复数上的仿射李代数，λ是零级整权，它是零级基本权π_i，i=1，.N，允许重复。我们考虑量子仿射代数U_{q}(G)上的量子Weyl模W_{q}(λ)，它对应于λ的某些Drinfeld多项式。已知W_(Q)(λ)的经典极限(即“q=1”极限)成为仿射李代数g上的Weyl模W(λ)，并且已知Weyl模W(λ)同构(作为对应于g的当前代数上的模)到Weyl模W(π_i)的融合积，i=1，.在我们以前的工作中，我们证明了Lakshmibai-Seshadri路径(模为g的零根λ)的晶体B(δ){CL}与量子λ模W_{q}(λ)的晶基同构。此外，我们还证明了晶体B(λ)_{CL}与晶体B(π_i)的张量积B同构，i=1，…，n。在2005年至2007年的一系列工作中，我们在上述晶体B(λ)_{CL}上定义了一个(非负)整值函数(我们称之为度函数)，并证明了这个度函数可以(通过B(λ)_{CL}与B的同构)与B上的“能量函数”同构，这源于统计力学中关于可解晶格模型的研究。特别地，通过将自己限制在A型仿射李代数的情况下，我们得到了Kostka多项式的Lakshmibai-Seshadri路的描述。

Let g be an affine Lie algebra over the complex numbers, and let λ be a level-zero integral weight that is a sum of level-zero fundamental weights π_I, I=1, . N, with repetitions allowed. We consider the quantum Weyl module W_{q} (λ) over the quantum affine algebra U_{q} (g) associated to certain Drinfeld polynomials corresponding to λ. It is known that the classical limit (I. e., "q=1" limit) of W_{q} (λ) becomes the Weyl module W (λ) over the affine Lie algebra g. Also, it is known that the Weyl module W (λ) is isomorphic (as a module over the Current algebra corresponding to g) to a fusion product of the Weyl modules W (π_I), I=1, . N.In our previous works, we showed that the crystal B (λ)_{cl} of Lakshmibai-Seshadri paths (modulo the null root δ of g) of shape λ is isomorphic as a crystal to the crystal basis of the quantum Weyl module W_{q} (λ). Moreover, we showed that the crystal B (λ)_{cl} is isomorphic as a crystal to a tensor product B of the crystals B (π_I), I=1, ., nIn our series of works from 2005 to 2007, we defined a certain (nonnegative) integer-valued function (which we call the degree function) on the crystal B (λ)_{cl} above, and proved that this degree function can be identified (through the isomorphism between B (λ)_{cl} and B) with the "energy function" on B, which arose from the study of solvable lattice models in statistical mechanics. In particular, by restricting ourselves to the case of affine Lie algebras of type A, we obtain a description of Kostka polynomials in terms of Lakshmibai-Seshadri paths.