Просмотр новости
Найдите то, что Вас интересует
Спутники
РН
SpaceX
Роскосмос
NASA
СпутникиРНSpaceXРоскосмосNASASpeaker: Mikhail MazinSpeaker Affiliation: Kansas StateOctober 20, 2026ESB 4133
CanadaView All EventsAbstract: The classical q,t-Catalan polynomials can be computed as enumerations of subpartitions of staircase partitions with respect to the area and dinv statistics. The triangular q,t-Catalan polynomials are a generalization where the staircase partitions are replaced with more general triangular partitions. The celebrated Shuffle Theorems provide a connection between the (triangular) q,t-Catalan polynomials and the action of the so-called Negut's elements of the elliptic Hall algebra on symmetric functions. In turn, this allows one to generalize the q,t-Catalan polynomials beyond the case of triangular partitions. However, the formula obtained from the elliptic Hall algebra action is not a positive sum of monomials, but rather a complicated sum of rational functions. Nevertheless, it is known that for any partition lambda one obtains a polynomial with integer coefficients, and its specialization at t=1 counts subpartitions of lambda with respect to area. Furthermore, computer computations suggest that if lambda is concave, then the coefficients of the polynomial are non-negative. A natural question is then to generalize the second statistic, dinv, to the case of concave partitions.
We propose a geometric approach to generalizing the dinv statistic. Given a concave partition lambda we consider the Hilbert scheme of points of the monomial ideal corresponding to lambda. These Hilbert schemes are highly singular. However, they are embedded into the Hilbert scheme of points on the complex plane. Intersecting with Bialynicki-Birula cells, one obtains a decomposition enumerated by subpartitions of lambda. With some luck, one can hope that the pieces of the decomposition are affine cells and then use their dimensions to generalize the dinv statistic. If lambda is triangular, then this construction recovers the usual dinv statistic. In addition, the construction provides a generalization of the dinv statistic for the following families of partitions lambda: 1) hooks; 2) partitions with at most three rows; 3) partitions (k,2,1,1,...), i.e. hooks with an extra box (1,1). (Case 3) was developed Jacob Bucciarelli in his PhD thesis.)
In this talk I will introduce the Negut's operators, Shuffle Theorems, and the generalized q,t-Catalan polynomials. I will then turn to the proposed geometric model: the Hilbert schemes of monomial ideals, focusing mostly on the three series of cases mentioned above.
Hilbert schemes of monomial ideals and generalized q,t-Catalan polynomialsEvent Topic:
Discrete mathematics
Speaker:
Mikhail Mazin
Speaker Affiliation:
Kansas State
October 20, 2026
ESB 4133
Canada
Abstract:
The classical q,t-Catalan polynomials can be computed as enumerations of subpartitions of staircase partitions with respect to the area and dinv statistics. The triangular q,t-Catalan polynomials are a generalization where the staircase partitions are replaced with more general triangular partitions. The celebrated Shuffle Theorems provide a connection between the (triangular) q,t-Catalan polynomials and the action of the so-called Negut's elements of the elliptic Hall algebra on symmetric functions. In turn, this allows one to generalize the q,t-Catalan polynomials beyond the case of triangular partitions. However, the formula obtained from the elliptic Hall algebra action is not a positive sum of monomials, but rather a complicated sum of rational functions. Nevertheless, it is known that for any partition lambda one obtains a polynomial with integer coefficients, and its specialization at t=1 counts subpartitions of lambda with respect to area. Furthermore, computer computations suggest that if lambda is concave, then the coefficients of the polynomial are non-negative. A natural question is then to generalize the second statistic, dinv, to the case of concave partitions.
We propose a geometric approach to generalizing the dinv statistic. Given a concave partition lambda we consider the Hilbert scheme of points of the monomial ideal corresponding to lambda. These Hilbert schemes are highly singular. However, they are embedded into the Hilbert scheme of points on the complex plane. Intersecting with Bialynicki-Birula cells, one obtains a decomposition enumerated by subpartitions of lambda. With some luck, one can hope that the pieces of the decomposition are affine cells and then use their dimensions to generalize the dinv statistic. If lambda is triangular, then this construction recovers the usual dinv statistic. In addition, the construction provides a generalization of the dinv statistic for the following families of partitions lambda: 1) hooks; 2) partitions with at most three rows; 3) partitions (k,2,1,1,...), i.e. hooks with an extra box (1,1). (Case 3) was developed Jacob Bucciarelli in his PhD thesis.)
In this talk I will introduce the Negut's operators, Shuffle Theorems, and the generalized q,t-Catalan polynomials. I will then turn to the proposed geometric model: the Hilbert schemes of monomial ideals, focusing mostly on the three series of cases mentioned above.
Event Topic:
| # | Наименование новости | Тональность | Информативность | Дата публикации |
|---|---|---|---|---|
| 1 | Quadratic forms and linear systems on abelian surfaces | 0 | 16.3 | 22-10-2026 |
| 2 | Triply-graded link homology and recursions | 0 | 21.25 | 23-10-2026 |
| 3 | Stably free modules without free summands | 0 | 15.47 | 23-10-2026 |
| 4 | Local Arthur packets: why we care, and how we compute them | 0 | 15.46 | 22-10-2026 |
| 5 | La conjecture d’André-Oort démontrée : un pas majeur dans la caractérisation de certaines variétés algébriques abstraites | 0 | 6.94 | 07-08-2026 |
| 6 | A New Model Formulation of the Spread of Epidemics: Why Were Traditional Models So Inaccurate for COVID-19? | 0 | 19.59 | 23-10-2026 |
| 7 | Musings on Constructions of Optimal Latin Hypercube Designs with Flexible Sizes | 0 | 6.96 | 22-07-2026 |
| 8 | Mulatu Polynomials and an Efficient Detection Algorithm for Mulatu Numbers [version 2; peer review: 2 not approved] | 0 | 10.55 | 11-08-2026 |
| 9 | CHEBYSHEV POLYNOMIALS ON REGULAR POLYGONS IN THE COMPLEX PLANE | 0 | 3.71 | 13-10-2026 |