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arXiv 1803.06754math.RTmath.QAmath.RA
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量子Grothendieck环同构、簇代数和Kazhdan-Lusztig算法

Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm

David Hernandez, Hironori Oya

备注：
v1 : 63 pages. v2 : 64 pages. Minor modification. To appear in Adv. Math
发表信息：
Adv. Math. 347 (2019), 192--272
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摘要：

建立了$B_n^{(1)}$型和$A_{2n-1}^{(1)}$型量子仿射代数有限维表示的某些显著幺半范畴的量子Grothendieck环之间的环同构.我们的证明部分依赖于相应的量子簇代数结构。此外，我们还证明了我们的同构在$t = 1$上是Kashiwara，Kim和Oh最近用其他方法得到的（经典）Grothendieck环的同构的特例.因此，我们证明了第一作者在2002年提出的一个猜想：对于类型$B_n^{(1)}$，上述范畴中标准模中的单模的重数是由Kazhdan-Lusztig多项式的某些类似物的特殊化给出的，并且这些多项式的系数是正的.

英文摘要：

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms specialize at $t = 1$ to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002 : the multiplicities of simple modules in standard modules in the categories above for type $B_n^{(1)}$ are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.

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