We study cyclotomic quiver Hecke algebras RΛ0(β) in type A2ℓ(2), where Λ0 is the fundamental weight. The algebras are natural A2ℓ(2)-type analogue of Iwahori–Hecke algebras associated with the ...symmetric group, from the viewpoint of the Fock space theory developed by the first author and his collaborators. We give a formula for the dimension of the algebra, and a simple criterion to tell the representation type. The criterion is a natural generalization of Erdmann and Nakanoʼs for the Iwahori–Hecke algebras. Except for the examples coming from cyclotomic Hecke algebras, no results of these kind existed for cyclotomic quiver Hecke algebras, and our results are the first instances beyond the case of cyclotomic Hecke algebras.
We are interested in the McKay quiver Γ(
G
) and skew group rings
A
∗
G
, where
G
is a finite subgroup of GL(
V
), where
V
is a finite dimensional vector space over a field
K
, and
A
is a
K
−
G
...-algebra. These skew group rings appear in Auslander’s version of the McKay correspondence. In the first part of this paper we consider complex reflection groups
G
⊆
GL
(
V
)
and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups
G
(
r
,
p
,
n
). We first look at the case
G
(1,1,
n
), which is isomorphic to the symmetric group
S
n
, followed by
G
(
r
,1,
n
) for
r
> 1. Then, using Clifford theory, we can determine the McKay quiver for any
G
(
r
,
p
,
n
) and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra
A
~
(
G
)
of a finite group
G
⊆
GL
(
V
)
, which is Morita equivalent to the skew group ring
A
∗
G
. This description gives us an embedding of the basic algebra Morita equivalent to
A
∗
G
into a matrix algebra over
A
.
For a certain class of partitions, a simple qualitative relation is observed between the shape of the Young diagram and the pattern of zeros of the Wronskian of the corresponding Hermite polynomials. ...In the case of the two-term Wronskian W(Hn,Hn+k), we give an explicit formula for the asymptotic shape of the zero set as n→∞. Some empirical asymptotic formulas are given for the zero sets of three-term and four-term Wronskians.
► We link the zero set of Wronskians of Hermite polynomials with the shape of Young diagrams. ► For the two-term Wronskian, an explicit formula for the asymptotic shape of the zero set is found. ► For three-term and four-term Wronskians, some empirical asymptotic formulas are given.
We introduce a large class of random Young diagrams which can be regarded as a natural one-parameter deformation of some classical Young diagram ensembles; a deformation which is related to
Jack ...polynomials
and
Jack characters
. We show that each such a random Young diagram converges asymptotically to some limit shape and that the fluctuations around the limit are asymptotically Gaussian.
We study root-theoretic Young diagrams to investigate the existence of a Lie-type uniform and nonnegative combinatorial rule for Schubert calculus.
We provide formulas for (co)adjoint varieties of ...classical Lie type. This is a simplest case after the (co)minuscule family (where a rule has been proved by H. Thomas and the second author using work of R. Proctor). Our results build on earlier Pieri-type rules of P. Pragacz–J. Ratajski and of A. Buch–A. Kresch–H. Tamvakis. Specifically, our formula for OG(2,2n) is the first complete rule for a case where diagrams are non-planar. Yet the formulas possess both uniform and non-uniform features.
Using these classical type rules, as well as results of P.-E. Chaput–N. Perrin in the exceptional types, we suggest a connection between polytopality of the set of nonzero Schubert structure constants and planarity of the diagrams. This is an addition to work of A. Klyachko and A. Knutson–T. Tao on the Grassmannian and of K. Purbhoo–F. Sottile on cominuscule varieties, where the diagrams are always planar.
In this paper we describe the spectrum of the quantum periodic Benjamin-Ono equation in terms of the multi-phase solutions of the underlying classical system (the periodic multi-solitons). To do so, ...we show that the semi-classical quantization of this system given by Abanov-Wiegmann is exact and equivalent to the geometric quantization by Nazarov-Sklyanin. First, for the Liouville integrable subsystems defined from the multi-phase solutions, we use a result of Gérard-Kappeler to prove that if one neglects the infinitely-many transverse directions in phase space, the regular Bohr-Sommerfeld conditions on the actions are equivalent to the condition that the singularities of the Dobrokhotov-Krichever multi-phase spectral curves define an anisotropic partition (Young diagram). Next, we locate the renormalization of the classical dispersion coefficient by Abanov-Wiegmann in the realization of Jack functions as quantum periodic Benjamin-Ono stationary states. Finally, we show that the classical energies of Bohr-Sommerfeld multi-phase solutions in the renormalized theory give the exact quantum spectrum found by Nazarov-Sklyanin without any Maslov index correction.
We develop a geometric approach to the study of $(s,ms-1)$-core and $(s,ms+1)$-core partitions through the associated $ms$-abaci. This perspective yields new proofs for results of H. Xiong and A. ...Straub on the enumeration of $(s, s+1)$ and $(s,ms-1)$-core partitions with distinct parts. It also enumerates $(s, ms+1)$-cores with distinct parts. Furthermore, we calculate the weight of the $(s, ms-1,ms+1)$-core partition with the largest number of parts. Finally we use 2-core partitions to enumerate self-conjugate core partitions with distinct parts. The central idea is that the $ms$-abaci of maximal $(s,ms\pm1)$-cores can be built up from $s$-abaci of $(s,s\pm 1)$-cores in an elegant way.
In this note we compute the Betti numbers for the Heisenberg Lie algebras. This is a well-known result due to Santharoubane. We re-derive this by considering Heisenberg Lie algebras as nilradicals of ...a certain subalgebra of a special linear Lie algebra and computing the dimensions of certain irreducible modules by representing them using Young diagrams.
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