In this paper we introduce a novel family of Markov chains on the simple representations of SL2$\left ({\mathbb {F}_p}\right )$
F
p
in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We show these chains are reversible and find their connected components and their stationary distributions. We draw connections between the properties of the chain and the representation theory of SL2$\left ({\mathbb {F}_p}\right )$
F
p
, emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple SL2$\left ({\mathbb {F}_p}\right )$
F
p
-representations.
This paper proves an identity between flagged Schur polynomials, giving a duality between row flags and column flags. This identity generalises both the binomial determinant duality theorem due to Gessel and Viennot and the symmetric function duality theorem due to Aitken. As corollaries we obtain the lifts of the binomial determinant duality theorem to \(q\)-binomial coefficients and to symmetric polynomials. Our method is a path counting argument on a novel lattice generalising that used by Gessel and Viennot.
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