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We categorify the Beilinson–Lusztig–MacPherson integral form of quantum sl ( 2 ) specialized at a prime root of unity. [ABSTRACT FROM AUTHOR]

We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan–Lusztig polynomial of M , 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 a q -deformation of the Möbius algebra of M , 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. [ABSTRACT FROM AUTHOR]

We compute the trace decategorification of the Hecke category for an arbitrary Coxeter group. More generally, we introduce the notion of a strictly object-adapted cellular category and calculate the trace for such categories. [ABSTRACT FROM AUTHOR]

We categorify an idempotented form of quantum sl 2 and some of its simple representations at a prime root of unity. [ABSTRACT FROM AUTHOR]