In the 1990's Garsia and Haiman introduced the diagonal coinvariant ring, which generalizes the classical coinvariant ring (the cohomology ring of the flag variety) to two sets of commuting variables. Since then, there has been much interest in studying generalizations with k sets of n commuting variables and j sets of n anticommuting variables, called a bosonic-fermionic coinvariant ring R_n^{(k,j)}, for various choices of small k and j. There is also an analogous story for the Weyl group of type B_n (the hyperoctahedral group); denote the type B bosonic-fermionic coinvariant ring by R_{B_n}^{(k,j)}.
In this talk, we will learn about the history of these families of coinvariant rings, paying attention to when the story is the same in both types A and B, and when they diverge. New results include an explicit formula for the GL(2) x B_n module structure of R_{B_n}^{(0,2)}, the sign character of R_{B_n}^{(0,3)}, and the standard characters in types A and B. This is based on joint work with Yuhan Jiang (UC Berkeley).