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Tail positive words and generalized coinvariant algebras. (English) Zbl 1369.05214

Summary: Let \(n,k,\) and \(r\) be nonnegative integers and let \(S_n\) be the symmetric group. We introduce a quotient \(R_{n,k,r}\) of the polynomial ring \(\mathbb{Q}[x_1, \dots, x_n]\) in \(n\) variables which carries the structure of a graded \(S_n\)-module.  When \(r \geq n\) or \(k = 0\) the quotient \(R_{n,k,r}\) reduces to the classical coinvariant algebra \(R_n\) attached to the symmetric group. Just as algebraic properties of \(R_n\) are controlled by combinatorial properties of permutations in \(S_n\), the algebra of \(R_{n,k,r}\) is controlled by the combinatorics of objects called tail positive words. We calculate the standard monomial basis of \(R_{n,k,r}\) and its graded \(S_n\)-isomorphism type. We also view \(R_{n,k,r}\) as a module over the 0-Hecke algebra \(H_n(0)\), prove that \(R_{n,k,r}\) is a projective 0-Hecke module, and calculate its quasisymmetric and nonsymmetric 0-Hecke characteristics. We conjecture a relationship between our quotient \(R_{n,k,r}\) and the delta operators of the theory of Macdonald polynomials.

MSC:

05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E05 Symmetric functions and generalizations
20C08 Hecke algebras and their representations
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References:

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