Mirabolic quantum \(sl(n)\) is an associative algebra defined by a convolution product on the space of triples of two partial flags and a vector. We show that it has the structure of a comodule algebra for the quantum enveloping algebra of \(sl(n)\), and use this to classify all of its finite dimensional representations. We also give an explicit description of the correspondence between these representations and the ones for the mirabolic Hecke algebra, which is given by a mirabolic quantum Schur-Weyl duality. This is joint work with Pallav Goyal (UC Riverside)..

Turyn-type sequences, TT(n), are quadruples of ±1-sequences X,Y, Z,W with lengths n,n,n,n −1 respectively, such that a certain sum of their respective nonperiodic autocorrelation functions is always equal to 0. Turyn-type sequences are known to exist for all even \(n \leq 38\). In this paper we construct the first examples of TT(40), TT(42), TT(44). Each one of these new Turyn-type sequences gives rise to an infinite series of Hadamard matrices among various other consequences.

Classically algebraic K-theory captures various invariants associated to a ring, R, used widely in algebraic and arithmetic geometry and number theory. To compute these invariants, one of the most successful tools have been of trace methods. The trace method machinery builds off of a map called the Dennis trace map from algebraic K-theory of R to Hochschild homology of R. In recent few years, new 'combinatorial' K-theories have been introduced (first by Zakharevich) motivated by generalized Hilbert's third problem. In this talk, we present initial attempts to generalize the theory of trace methods to such combinatorial K-theories.

We introduce a new broadly unifying family of combinatorial objects, which we call permutation flows, associated to an acyclic directed graph \( G \) together with a framing, which is a collection of total orders on the sets of incoming and outgoing edges at each vertex. This new family is combinatorially rich: containing as special cases various families of combinatorial objects that are frequently studied in the literature, including permutations, circular permutations, multipermutations, Stirling permutations, Catalan objects and their generalizations. When permutation flows are decorated with shuffles satisfying a compatibility condition, they also include the combinatorics of parking functions and their generalizations.

This model is geometrically rich: permutation flow shuffles define a family of unimodular triangulations of the flow polytope \( \mathcal{F}_F(\mathbf{a}) \) on \( G \) with an integer balanced netflow vector \( \mathbf{a} \) where only the last entry is negative. Permutation flow triangulations extend the Danilov, Karzanov, and Koshevoy triangulations that were defined for the case where \( \mathbf{a} = \mathbf{e}_0 - \mathbf{e}_n \). We show that the \( h^* \)-polynomial of \( \mathcal{F}_F(\mathbf{e}_0 - \mathbf{e}_n) \) is the \( G \)-Eulerian polynomial.

The model comes with an order structure induced by intuitive operators on permutation flows which we call the weak order. This order includes as special cases the weak order on permutations, the Tamari lattice, order ideals in Young’s lattice, and their generalizations.

Joint work with C. R. H. Hanusa, A. H. Morales, and M. Yip.

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