The projective coinvariant algebra

Abstract: The coinvariant algebra, the quotient of the coordinate ring of (A^1)^n=A^n by the ideal generated by positive degree invariant polynomials, plays a basic role in algebraic combinatorics and the representation theory of the symmetric group Sn, equipping its regular representation with a graded algebra structure. Using the coordinate ring of (P^1)^n in its Segre embedding, I will introduce a degeneration of the coinvariant algebra, the projective coinvariant algebra,which gives a bigraded structure on the regular representation of S_n with interesting Frobenius character that generalises a classical result of Lusztig and Stanley. I will also show how this algebra contains bigraded versions of partial coinvariant algebras, coming from coordinate rings of all possible Segre embeddings. Relations to Haiman’s diagonal coinvariant algebra, and a certain equivariant Hilbert scheme, will also be discussed.