q, t-Catalan numbers and generators for the radical ideal defining the diagonal locus of (C)
نویسندگان
چکیده
Let I be the ideal generated by alternating polynomials in two sets of n variables. Haiman proved that the q, t-Catalan number is the Hilbert series of the bi-graded vector space M(= ⊕ d1,d2 Md1,d2) spanned by a minimal set of generators for I. In this paper we give simple upper bounds on dim Md1,d2 in terms of number of partitions, and find all bi-degrees (d1, d2) such that dimMd1,d2 achieve the upper bounds. For such bi-degrees, we also find explicit bases for Md1,d2 .
منابع مشابه
Notes on a Minimal Set of Generators for the Radical Ideal Defining the Diagonal Locus of (c)
We provide explicit generators for the radical ideal defining the diagonal locus of (C) of certain bi-degrees. As a consequence, we discover a relation between t, q-Catalan numbers and partition numbers.
متن کاملq,t-Catalan Numbers and Generators for the Radical Ideal defining the Diagonal Locus of (C2)n
Let I be the ideal generated by alternating polynomials in two sets of n variables. Haiman proved that the q, t-Catalan number is the Hilbert series of the graded vector space M(= ⊕ d1,d2 Md1,d2) spanned by a minimal set of generators for I. In this paper we give simple upper bounds on dim Md1,d2 in terms of partition numbers, and find all bi-degrees (d1, d2) such that dimMd1,d2 achieve the upp...
متن کاملOn the diagonal ideal of (C) and q, t-Catalan numbers
Let In be the (big) diagonal ideal of (C). Haiman proved that the q, t-Catalan number is the Hilbert series of the graded vector space Mn = ⊕ d1,d2 (Mn)d1,d2 spanned by a minimal set of generators for In. We give simple upper bounds on dim (Mn)d1,d2 in terms of partition numbers, and find all bi-degrees (d1, d2) such that dim (Mn)d1,d2 achieve the upper bounds. For such bi-degrees, we also find...
متن کاملSQUARE q , t - LATTICE PATHS AND ∇ ( p n ) NICHOLAS A
The combinatorial q, t-Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The q, t-Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the n’th q, t-Catalan number is the Hilb...
متن کاملSQUARE q , t - LATTICE PATHS AND ∇ ( p n ) NICHOLAS
The combinatorial q, t-Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The q, t-Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the n’th q, t-Catalan number is the Hilb...
متن کامل