Quiver Schur algebras and q-Fock space
Quiver Schur algebras and q-Fock space
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发表时间：
2011-10
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通讯作者：
C. Stroppel;Ben Webster
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作者：
C. Stroppel;Ben Webster
在Brundan-Kleshchev关于Hecke代数和Ariki关于Q-Schur代数的工作的精神下，我们发展了分圆Q-Schur代数理论的一个分次版本。作为应用，我们用分圆Q-Schur代数的分解数的q-模拟来确定高阶Fock空间上的标准基的系数。我们将分圆Q-Schur代数表示为产生于箭图几何中的卷积代数的商，因此我们称之为箭图Schur代数。这些代数也被图解表示，在风格上类似于最近Khovanov和Lauda的构造。它们也是明显分次的，因此为分圆Q-Schur代数配备了一个不明显的分次。在此基础上，我们构造了该代数的一个分次细胞基，类似于Mathas，Hu，Brundan和第一作者关于割圆Hecke代数的构造。从更高表示理论的角度来看，箭图Schur代数也很有趣。已知某些割圆商的Grothendieck群的和与更高层次的Fock空间一致。我们证明了我们的分次版本定义了一个更高的q-Fock空间(定义为1级q-变形Fock空间的张量积)。在此基础上，给出了不可分解投射模的标准基和Weyl模的标准基。这使得我们可以证明已经描述的分解数和典范基之间的关系。
We develop a graded version of the theory of cyclotomic q-Schur algebras, in the spirit of the work of Brundan-Kleshchev on Hecke algebras and of Ariki on q-Schur algebras. As an application, we identify the coefficients of the canonical basis on a higher level Fock space with q-analogues of the decomposition numbers of cyclotomic q-Schur algebras. We present cyclotomic q-Schur algebras as a quotient of a convolution algebra arising in the geometry of quivers; hence we call these quiver Schur algebras. These algebras are also presented diagrammatically, similar in flavor to a recent construction of Khovanov and Lauda. They are also manifestly graded and so equip the cyclotomic q-Schur algebra with a non-obvious grading. On the way we construct a graded cellular basis of this algebra, resembling the constructions for cyclotomic Hecke algebras by Mathas, Hu, Brundan and the first author. The quiver Schur algebra is also interesting from the perspective of higher representation theory. The sum of Grothendieck groups of certain cyclotomic quotients is known to agree with a higher level Fock space. We show that our graded version defines a higher q-Fock space (defined as a tensor product of level 1 q-deformed Fock spaces). Under this identification, the indecomposable projective modules are identified with the canonical basis and the Weyl modules with the standard basis. This allows us to prove the already described relation between decomposition numbers and canonical bases.