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.

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