Generalized coinvariant algebras for wreath products

Chan, Kin Tung Jonathan; Rhoades, Brendon

doi:10.1016/j.aam.2020.102060

Cited by 3 publications

(8 citation statements)

References 16 publications

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“…In Lemma 4 below we will show that when r = 1 this notion of comaj coincides with the comaj-statistic defined in Haglund, Rhoades and Shimozono [7]. However, for r > 1 this notion of comaj is not complementary to the maj-statistic defined in Chan and Rhoades [2], as they use a descent-starred model for ordered set partitions. Furthermore, their comaj-statistic includes a sum over the complements of the colors of the ordered set partition, whereas our definition includes simply a sum over the colors of the ordered set partition.…”

mentioning

confidence: 86%

“…In the above example, we would have g = 4 3 2 2 3 2 9 1 6 1 1 0 5 2 7 2 8 1 , which has Des(g) = {4, 6}, so des(g) = 2 and of the 6 ascents, the first, second, fourth and sixth are starred. Representing this as a path from (0, 2) to (2,4), where each starred ascent contributes a step east and each unstarred ascent contributes a step south, we get the following picture…”

Section: Ordered Set Partitions and G N Facesmentioning

confidence: 99%

“…For example, when n = 6, k = 3 and r = 2, the partitions (5, 5, 2, 2, 2), (6, 5, 5, 5, 1), (6, 5, 4, 4, 2, 2, 2, 1) and (6,6,2) are admissable, semi-admissable, non-admissable and nonadmissable respectively, both when (A, A) = (S n,k , S n,k ) and when (A, A) = (R n,k , R n,k ). However, the partition (6, 5, 5, 2, 2, 2) is non-admissable if (A, A) = (S 6,3 , S 6,3 ), but admissable for (R 6,3 , R 6,3 ).…”

Section: Each Variable Occurring Inb (Gλ)mentioning

confidence: 99%

“…Furthermore, when r > 1 Chan and Rhoades [2] define two generalizations of the coinvariant algebra attached to G(r, 1, n). Similarly to the case r = 1, one of these modules is isomorphic to C[OP n,k is the set of k-dimensional faces attached to the Coxeter complex of G(r, 1, n) which can be thought of as r-colored ordered set partitions of some set S ⊆ [n] into k blocks.…”

Section: Introductionmentioning

confidence: 99%

“…In [2], Chan and Rhoades define two generalizations of R n for all pairs of integers 0 k n with n 1.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Generalized Coinvariant Algebras for $G(r,1,n)$ in the Stanley-Reisner setting

Kroes

2019

Electron. J. Combin.

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Let $r$ and $n$ be positive integers, let $G_n$ be the complex reflection group of $n \times n$ monomial matrices whose entries are $r^{\textrm{th}}$ roots of unity and let $0 \leq k \leq n$ be an integer. Recently, Haglund, Rhoades and Shimozono ($r=1$) and Chan and Rhoades ($r>1$) introduced quotients $R_{n,k}$ (for $r>1$) and $S_{n,k}$ (for $r \geq 1$) of the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ in $n$ variables, which for $k=n$ reduce to the classical coinvariant algebra attached to $G_n$. When $n=k$ and $r=1$, Garsia and Stanton exhibited a quotient of $\mathbb{C}[\mathbf{y}_S]$ isomorphic to the coinvariant algebra, where $\mathbb{C}[\mathbf{y}_S]$ is the polynomial ring in $2^n-1$ variables whose variables are indexed by nonempty subsets $S \subseteq [n]$. In this paper, we will define analogous quotients that are isomorphic to $R_{n,k}$ and $S_{n,k}$.

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mentioning

confidence: 86%

Section: Ordered Set Partitions and G N Facesmentioning

confidence: 99%

Section: Each Variable Occurring Inb (Gλ)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In [2], Chan and Rhoades define two generalizations of R n for all pairs of integers 0 k n with n 1.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Generalized Coinvariant Algebras for $G(r,1,n)$ in the Stanley-Reisner setting

Kroes

2019

Electron. J. Combin.

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q-Stirling numbers in type B

Sagan,

Swanson

2024

European Journal of Combinatorics

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Orbit harmonics for the union of two orbits

T.¹

2022

Preprint

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Garsia and Procesi, in their study of Springer's representation, proved that the cohomology ring of a Springer fiber is isomorphic to the associated graded ring of the coordinate ring of the S n orbit of a single point in C n . This construction was an essential tool in their analysis of the Springer representation, and variations of it have reappeared recently in several other combinatorial and geometric contexts under the name orbit harmonics. In this article, we analyze the orbit harmonics of a union of two S n orbits. We prove that when the coordinate sums of the two orbits are different, the corresponding graded S n representation is a direct sum of two Springer representations, one of which is shifted in degree by 1.

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Generalized coinvariant algebras for wreath products

Cited by 3 publications

References 16 publications

Generalized Coinvariant Algebras for $G(r,1,n)$ in the Stanley-Reisner setting

Generalized Coinvariant Algebras for $G(r,1,n)$ in the Stanley-Reisner setting

q-Stirling numbers in type B

Orbit harmonics for the union of two orbits

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