On the quotient ring by diagonal invariants

By I. Gordon. 2002

131 citations recorded by the source corpus.

For a finite Coxeter group, W, and its reflection representation 𝔥, we find the character and Hilbert series for a quotient ring of ℂ[𝔥⊕𝔥*] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2e0b000160a972b415e2803770aefaecb4c9dcab

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Catalan numbers for complex reflection groups (2009, result): “We now prove our main result, answering a question of the second author, [7, Section 8], and giving a general and case-free construction of the Koszul resolutions produced in [1, 5, 7, 16].”
Cyclic Sieving of Noncrossing Partitions for Complex Reflection Groups (2007, background): “For notational purposes, recall that if σ lies in the Galois group Gal(Q[ζ|W |]/Q), one can form the Galois conjugate representation Uσ in which an element w ∈ W acting on U by a given matrix A(w) will act by the matrix A(w)σ on Uσ.”
Parking Spaces (2012, methodology): “For example, this quotient occurs in the rational Cherednik theory [27, 9, 20, 21] as the finite-dimensional irreducible module L(h+1)/h(1W ) corresponding to the trivial representation 1W of W , and taken at the rational parameter h+1 h .”
FI_W-modules and stability criteria for representations of the classical Weyl groups (2013, background): “The r-diagonal coinvariant algebras have been studied since the 1990s, with major contributors including Garsia, Haiman, Hagland, Gordon, Bergeron, and Biagioli; see Section 7.2 for more history.”
Combinatorics, symmetric functions, and Hilbert schemes (2002, background): “Gordon [34] has proved a conjecture of mine along these lines.”
Group actions on algebras and the graded Lie structure of Hochschild cohomology (2009, methodology): “Gordon [11] used these algebras to prove a version of the n! conjecture for Weyl groups due to Haiman.”
Multivariate diagonal coinvariant spaces for complex reflection groups (2011, background): “To better see what is striking here, it is worth recalling that , although the case r = 1 has a long history [4, 13, 14], it is only recently that the special case r = 2 has been somewhat settled [3, 6, 7, 10].”
Orthogonal functions generalizing Jack polynomials (2007, methodology): “The paper (16) uses the representation theory of rational Cherednik algebras to prove a conjecture of Haiman on the ring of diagonal coinvariants of a Weyl group W .”
Higher Trivariate Diagonal Harmonics via generalized Tamari Posets (2011, background): “If we specialize one of the parameters (say q3) to 0, we get back the graded Hilbert series of bivariate diagonal harmonics (of overall dimension (n + 1)n−1) which has received a lot of attention in recent years (see [8, 10, 13, 14, 15, 18, 21]).”
Representations of Rational Cherednik algebras with minimal support and torus knots (2013, methodology): “ecial case when Wis a real irreducible reﬂection group, c= m/h, where his the Coxeter number of W, m∈Z≥1, GCD(m,h) = 1, and V = his the reﬂection representation, this complex was studied in [BEG] and [Go]. In this case, this complex is actually a resolution of a ﬁnite dimensional Hc(W,h)-module of d”
Finite-dimensional representations of rational Cherednik algebras (2002, methodology): “Remark. After the bulk of this paper had been written, we received a preliminary version of a very interesting preprint by I. Gordon [Go], containing results essentially equivalent to our Theorems 1.4(i), and 1.6, 1.10, and 2.3 (for the case c = 1 + 1 h). The main ideas of [ Go ] are very similar to”
Refined knot invariants and Hilbert schemes (2013, unknown): “3 ([26]).”