1,721,011 research outputs found
DEGENERATE SCHUBERT VARIETIES IN TYPE A
No full textWe introduce rectangular elements in the symmetric group. In the framework of PBW degenerations, we show that in type A the degenerate Schubert variety associated with a rectangular element is indeed a Schubert variety in a partial ag variety of the same type with larger rank. Moreover, the degenerate Demazure module associated with a rectangular element is isomorphic to the Demazure module for this particular Schubert variety of larger rank. This generalises previous results by Cerulli Irelli, Lanini and Littelmann for the PBW degenerate ag variety in [CLL]
PBW-degenerated Demazure modules and Schubert varieties for triangular elements
Full text linkWe study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for sln+1. We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements. These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties
PBW-degenerated Demazure modules and Schubert varieties for triangular elements
No full textWe study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for sln+1. We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements. These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties
Extended partial order and applications to tensor products
No full textWe extend the preorder on k-tuples of dominant weights of a simple
complex Lie algebra g of classical type adding up to a fixed weight λ
defined by Chari, Sagaki and the author [Posets, tensor products and
Schur positivity, Algebra and Number Theory, to appear]. We show that
the induced extended partial order on the equivalence classes has a unique
minimal and a unique maximal element. For k = 2 we compute its size
and determine the cover relation.<p></p>
To each k-tuple we associate a tensor product of simple g-modules
and we show that for k = 2 the dimension increases also along with
the extended partial order, generalizing a theorem proved in the aforementioned
paper. We also show that the tensor product associated to the
maximal element has the biggest dimension among all tuples for arbitrary
k, indicating that this might be a symplectic (respectively, orthogonal)
analogue of the row shuffle defined by Fomin et al. [Amer. J. Math. 127
(2005), 101–127].<p></p>
The extension of the partial order reduces the number elements in
the cover relation and may facilitate the proof of an analogue of Schur
positivity along the partial order for symplectic and orthogonal types.<p></p>
Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence
No full textWe analyze marked poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appear naturally in the representation theory of semi-simple Lie algebras, and hence we can give a necessary and sufficient condition on the marked poset such that the associated toric degenerations of the corresponding partial flag variety are isomorphic. We further show that the set of lattice points in such a marked poset polytope is the Minkowski sum of sets of lattice points for 0–1 polytopes. Moreover, we provide a decomposition of the marked poset into indecomposable marked posets, which respects this Minkowski sum decomposition for the marked chain polytopes
New homogeneous ideals for current algebras: filtrations, fusion products and Pieri rules
No full textNew graded modules for the current algebra of sln are introduced. Relating these modules to the fusion product of simple sln-modules and local Weyl modules of truncated current algebras shows their expected impact on several outstanding conjectures. We further generalize results on PBW filtrations of simple sln-modules and use them to provide decomposition formulas for these new modules in important cases
New homogeneous ideals for current algebras: filtrations, fusion products and Pieri rules
No full textNew graded modules for the current algebra of sln are introduced. Relating these modules to the fusion product of simple sln-modules and local Weyl modules of truncated current algebras shows their expected impact on several outstanding conjectures. We further generalize results on PBW filtrations of simple sln-modules and use them to provide decomposition formulas for these new modules in important cases
Some results on finite dimensional modules of current and loop algebras
No full textIn the field of finite dimensional modules of current and loop algebras a lot of research was done and progress was made in the last two decades. Resuming the discussions we showed in the present thesis that Demazure modules are fusion products of ``smaller`` Demazure modules and calculated a decomposition of the fundamental Demazure modules as g-modules. Combining both results we obtained new dimension and character formulas for Demazure modules. As an application we constructed certain affine highest weight modules as the direct limit of fusion products of Demazure modules. We proved that the fundamental Demazure modules are isomorphic to Kirillov-Reshetikhin modules for the current algebra. Furthermore we proved as an analogon in the combinatorial representation theory that the Kirillov-Reshetikhin crystal contains the Demazure crystal. We give a new and elementary proof of the dimension formula of Weyl modules for the loop algebra in the simply laced case. For twisted loop algebras we provide a classification of Weyl modules and proved the analog theorems from the untwisted case, e.g. tensor product structure, dimension and character formulas
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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