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，设λ是一个零级积分权值，它是零级基本权值π_I， I=1，的和。N，允许重复。我们考虑量子Weyl模W_{q} （λ）在量子仿射代数U_{q} (g)上与λ对应的某些德林菲尔德多项式相关联。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）是Weyl模W (π_I)， I=1的融合积的同构（作为对应于g的当前代数上的模）。在我们之前的工作中，我们证明了形状为λ的Lakshmibai-Seshadri路径（模g的零根δ）的晶体B (λ)_{cl}与量子Weyl模W_{q} （λ）的晶体基是同构的。此外，我们还证明了晶体B (λ)_{cl}与晶体B (π_I)， I=1的张量积B是同构的。在我们2005 - 2007年的一系列工作中，我们在上面的晶体B (λ)_{cl}上定义了一个（非负的）整数值函数（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之间的同构）与B上的“能量函数”识别，这是在统计力学中可解晶格模型的研究中产生的。特别地，通过限制我们对A型仿射李代数的情况，我们得到了Lakshmibai-Seshadri路径下的Kostka多项式的描述。

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.