Diagrams of Hermitian type, highest weight modules, and syzygies of determinantal varieties

We study projective dimension and graded length of structural modules in parabolic-singular blocks of the BGG category O. Some of these are calculated explicitly, others are expressed in terms of two functions. We also obtain several partial results and estimates for these two functions and relate them to monotonicity properties for quasi-hereditary algebras. The results are then applied to study blocks of O in the context of Guichardet categories, in particular, we show that blocks of O are not always weakly Guichardet.MSC 2010 : 16E30, 17B10

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“…We will use freely that the Bruhat order on X µ is generated by right multiplication with simple reflections, see e.g. [EHP,Corollary 3.12].…”

Section: Hermitian Symmetric Pairsmentioning

confidence: 99%

Some homological properties of category 𝒪. IV

Coulembier

Mazorchuk

2016

Forum Mathematicum

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“…The Hasse diagram W p for the standard parabolic subalgebra p = p max α with α ∈ Π we can determine by the methods described at the end of Section 1.3.1. However, since p is a maximal parabolic subalgebra of g, which in other words means that the corresponding nilradical u is commutative, there is a relatively easy way how to compute the Hasse diagram W p , see [EHP14] for more details.…”

Section: Admissible Levels With Small Denominatormentioning

confidence: 99%

Admissible representations of simple affine vertex algebras

Futorny¹,

Morales²,

Křižka³

2021

Preprint

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We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra Lκ(sln+1) with n ∈ N of admissible level κ. For admissible simple highest weight modules corresponding to the principal, subregular and maximal parabolic nilpotent orbits we give a realization using the Gelfand-Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible sl2-induced modules in these orbits. In particular, simple admissible sl2-induced modules are fully classified for the principal nilpotent orbit.

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“…We let ‫ބ‬ ≤k ( p, q) denote the set of p by q matrices with rank at most k, which is a closed affine algebraic set, called a determinantal variety. For a relatively recent expository article about the role these varieties play in algebraic geometry and representation theory see [4]. In this section we consider the k = 1 case, in our context.…”

Section: 3mentioning

confidence: 99%

Expected value of the one-dimensional earth mover’s distance

Bourn¹,

Willenbring²

2020

Alg. Stat.

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From a combinatorial point of view, we consider the earth mover's distance (EMD) associated with a metric measure space. The specific case considered is deceptively simple: Let the finite set of integers [n] = {1,. .. , n} be regarded as a metric space by restricting the usual Euclidean distance on the real numbers. The EMD is defined on ordered pairs of probability distributions on [n]. We provide an easy method to compute a generating function encoding the values of EMD in its coefficients, which is related to the Segre embedding from projective algebraic geometry. As an application we use the generating function to compute the expected value of EMD in this one-dimensional case. The EMD is then used in clustering analysis for a specific data set.

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Diagrams of Hermitian type, highest weight modules, and syzygies of determinantal varieties

Cited by 14 publications

References 39 publications

Some homological properties of category 𝒪. IV

Some homological properties of category 𝒪. IV

Admissible representations of simple affine vertex algebras

Expected value of the one-dimensional earth mover’s distance

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