The Kazhdan-Lusztig polynomial of a matroid
The Kazhdan-Lusztig polynomial of a matroid
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DOI：
10.1016/j.aim.2016.05.005
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发表时间：
2014-12
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影响因子：
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通讯作者：
Ben Elias;N. Proudfoot;Max Wakefield
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文献类型：
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作者：
Ben Elias;N. Proudfoot;Max Wakefield
我们将每个拟阵Ma多项式与整数系数相关联，我们称之为M的Kazhdan-Lusztig多项式，类似于表示论中的Kazhdan-Lusztig多项式。我们猜想，系数总是非负的，我们证明了这一猜想的可表示拟阵解释我们的多项式作为交叉上同调庞加莱多项式。我们还引入了M的Möbius代数的aq-变形，并使用我们的多项式来定义这种变形的特殊基，类似于Hecke代数的标准基。我们猜想，在这个特殊的基础上乘法的结构系数是非负的，我们验证了这一猜想在许多例子。
We associate to every matroidMa polynomial with integer coefficients, which we call the Kazhdan–Lusztig polynomial ofM, in analogy with Kazhdan–Lusztig polynomials in representation theory. We conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincaré polynomials. We also introduce aq-deformation of the Möbius algebra ofM, and use our polynomials to define a special basis for this deformation, analogous to the canonical basis of the Hecke algebra. We conjecture that the structure coefficients for multiplication in this special basis are non-negative, and we verify this conjecture in numerous examples.