Tail Positive Words and Generalized Coinvariant Algebras

Let n, k, and r be nonnegative integers and let Sn be the symmetric group. We introduce a quotient Rn,k,r of the polynomial ring Q[x1, . . . , xn] in n variables which carries the structure of a graded Sn-module. When r > n or k = 0 the quotient Rn,k,r reduces to the classical coinvariant algebra Rn attached to the symmetric group. Just as algebraic properties of Rn are controlled by combinatorial properties of permutations in Sn, the algebra of Rn,k,r is controlled by the combinatorics of objects called tail positive words. We calculate the standard monomial basis of Rn,k,r and its graded Sn-isomorphism type. We also view Rn,k,r as a module over the 0Hecke algebra Hn(0), prove that Rn,k,r is a projective 0-Hecke module, and calculate its quasisymmetric and noncommutative 0-Hecke characteristics. We conjecture a relationship between our quotient Rn,k,r and the delta operators of the theory of Macdonald polynomials.