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One of the main open problems in Algebraic Combinatorics is to obtain a combinatorial interpretation for the Kronecker coefficients. The Kronecker coefficients, introduced by Murnaghan in 1938, are the multiplicities of irreducible representation in the decomposition of the tensor product of two irreducible representations of the symmetric group. As such they are naturally nonnegative numbers, yet to this day we have no positive combinatorial formula. 

Motivated by the Saxl conjecture and the tensor square conjecture, which conjectures that the tensor squares of certain irreducible representations of the symmetric group contain all irreducible representations, we study the tensor squares of irreducible representations associated with square Young diagrams. In this talk, I will discuss our work on the positivity of Kronecker coefficients for certain families of partitions and demonstrate various tools that can be used to compute Kronecker coefficients. I will also discuss the Newton polytope of the Kronecker product of Schur polynomials (this is joint work with Greta Panova).