Description
The coinvariant algebra, the quotient of the coordinate ring of affine n-space by the ideal generated by positive degree invariant polynomials, plays a basic role in algebraic combinatorics and the representation theory of the symmetric group S_n, equipping its regular representation with a graded algebra structure. Using the coordinate ring of the self-product of the projective line in its Segre embedding, I will introduce a degeneration of the coinvariant algebra, the projective coinvariant algebra. This gives a bigraded structure on the regular representation of S_n with interesting Frobenius character, generalising 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 corresponding to partitions of n. Finally I discuss a conjectured degeneration of the Haglund-Rhoades-Shimozono generalised coinvariant algebra.