The inverse Kazhdan-Lusztig polynomial of a matroid
The inverse Kazhdan-Lusztig polynomial of a matroid
复制标题
拟阵的 Kazhdan-Lusztig 反多项式
DOI：
10.1016/j.jctb.2021.07.004
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Xie Matthew H. Y.
中科院分区：
文献类型：
--
作者：
Gao Alice L. L.;Xie Matthew H. Y.
与经典的考西特群的Kazhdan-Lusztig多项式类似，Elias，Proudfoot和韦克菲尔德引入了拟阵的Kazhdan-Lusztig多项式的概念。经典的Kazhdan-Lusztig多项式和拟阵Kazhdan-Lusztig多项式都可以看作是Kazhdan-Lusztig-Stanley多项式在局部有限偏序集上的特殊情况。在Kazhdan-Lusztig-Stanley多项式的框架下，研究了Kazhdan-Lusztig-Stanley函数的逆，定义了拟阵的逆Kazhdan-Lusztig多项式.我们还计算这些多项式的布尔拟阵和一致拟阵。作为逆Kazhdan-Lusztig多项式的一个意外应用，我们得到了一个计算一致拟阵的Kazhdan-Lusztig多项式的新公式。类似于拟阵的Kazhdan-Lusztig多项式，我们猜想它的逆Kazhdan-Lusztig多项式的系数是非负的和对数凹的。
In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced the concept of Kazhdan-Lusztig polynomials for matroids. It is known that both the classical Kazhdan-Lusztig polynomials and the matroid Kazhdan-Lusztig polynomials can be considered as special cases of the Kazhdan-Lusztig-Stanley polynomials for locally finite posets. In the framework of Kazhdan-Lusztig-Stanley polynomials, we study the inverse of Kazhdan-Lusztig-Stanley functions and define the inverse Kazhdan-Lusztig polynomials for matroids. We also compute these polynomials for boolean matroids and uniform matroids. As an unexpected application of the inverse Kazhdan-Lusztig polynomials, we obtain a new formula to compute the Kazhdan-Lusztig polynomials for uniform matroids. Similar to the Kazhdan-Lusztig polynomial of a matroid, we conjecture that the coefficients of its inverse Kazhdan-Lusztig polynomial are nonnegative and log-concave.