Abstract
Let $\rho$ be a non-negative integer. A $\rho$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $\rho$ disjoint bases. We present a combinatorial formula for Kazhdan–Lusztig polynomials of $\rho$-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.
Highlights
The Kazhdan–Lusztig polynomial of a matroid was introduced by Elias, Proudfoot, and Wakefield in 2016 [2], which we define here
We instead prove the statement with our change of coordinates, that is, we show cia−1,b+2i−1 = #SkY T (a, i, b)
It will be helpful to restate this proposition in the following way for when we prove Theorem 11
Summary
The Kazhdan–Lusztig polynomial of a matroid was introduced by Elias, Proudfoot, and Wakefield in 2016 [2], which we define here. Note that when ρ = 0, the matroid Um,d(ρ) is Um,d, the uniform matroid of rank d on m + d elements In this case, Theorem 1 gives a new manifestly positive integral formula for the coefficients corresponding to the uniform matroid, which is the number of all legal fillings on the diagram above. This is because the bottom entry of the right-most column is always greater than d. Let cim,d be the ith coefficient of the Kazhdan-Lusztig Polynomial for the uniform matroid of rank d on m + d elements. It is still worth mentioning Theorem 2 as it can be proved using a lot less work
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