scholarly journals Combinatorial Study of Dellac Configurations and $q$-Extended Normalized Median Genocchi Numbers

10.37236/4068 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Ange Bigeni
Keyword(s):  
Flag Varieties ◽  
Combinatorial Interpretation ◽  
Genocchi Numbers ◽  
Degenerate Flag Varieties

In two recent papers, Feigin proved that the Poincaré polynomials of the degenerate flag varieties have a combinatorial interpretation through Dellac configurations, and related them to the $q$-extended normalized median Genocchi numbers $\bar{c}_n(q)$ introduced by Han and Zeng, mainly by geometric considerations. In this paper, we give combinatorial proofs of these results by constructing statistic-preserving bijections between Dellac configurations and two other combinatorial models of $\bar{c}_n(q)$.

scholarly journals Degenerate flag varieties and Schubert varieties: a characteristic free approach

2016 ◽  
Vol 284 (2) ◽  
pp. 283-308 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Martina Lanini ◽  
Peter Littelmann
Keyword(s):  
Schubert Varieties ◽  
Flag Varieties ◽  
Degenerate Flag Varieties

scholarly journals Degenerate flag varieties in network coding

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ghislain Fourier ◽  
Gabriele Nebe
Keyword(s):  
Network Coding ◽  
Affine Space ◽  
Linear Subspace ◽  
Flag Varieties ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Rank Metric ◽  
Rank Metric Codes ◽  
Information Set ◽  
Degenerate Flag Varieties

<p style='text-indent:20px;'>Building upon the application of flags to network coding introduced in [<xref ref-type="bibr" rid="b6">6</xref>], we develop a variant of this coding technique that uses degenerate flags. The information set is a metric affine space isometric to the space of upper triangular matrices endowed with the flag rank metric. This suggests the development of a theory for flag rank metric codes in analogy to the rank metric codes used in linear subspace coding.</p>

scholarly journals Quiver Grassmannians and degenerate flag varieties

2012 ◽  
Vol 6 (1) ◽  
pp. 165-194 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Evgeny Feigin ◽  
Markus Reineke
Keyword(s):  
Flag Varieties ◽  
Quiver Grassmannians ◽  
Degenerate Flag Varieties

scholarly journals Combinatorial Interpretations of the Kreweras Triangle in Terms of Subset Tuples

10.37236/7531 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Ange Bigeni
Keyword(s):  
Combinatorial Interpretation ◽  
Genocchi Numbers ◽  
Combinatorial Interpretations ◽  
Dumont Permutations

We show how the combinatorial interpretation of the normalized median Genocchi numbers in terms of multiset tuples, defined by Hetyei in his study of the alternation acyclic tournaments, is bijectively equivalent to previous models like the normalized Dumont permutations or the Dellac configurations, and we extend the interpretation to the Kreweras triangle.

scholarly journals Symplectic Degenerate Flag Varieties

2014 ◽  
Vol 66 (6) ◽  
pp. 1250-1286 ◽  
Author(s):  
Evgeny Feigin ◽  
Michael Finkelberg ◽  
Peter Littelmann
Keyword(s):  
Borel Subgroup ◽  
Classical Case ◽  
Explicit Construction ◽  
Flag Variety ◽  
Unipotent Group ◽  
Flag Varieties ◽  
Highest Weight ◽  
Weil Theorem ◽  
Degenerate Flag Varieties ◽  
Pbw Filtration

AbstractA simple finite dimensional module Vλ of a simple complex algebraic group G is naturally endowed with a filtration induced by the PBW-filtration of U(Lie G). The associated graded space is a module for the group Ga, which can be roughly described as a semi-direct product of a Borel subgroup of G and a large commutative unipotent group . In analogy to the flag variety ℱλ = G:[vλ] ⊂ ℙ(Vλ), we call the closure of the Ga-orbit through the highest weight line the degenerate flag variety . In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of G being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where, even for fundamental weights ω, the varieties differ from Fω. We give an explicit construction of the varieties and construct desingularizations, similar to the Bott–Samelson resolutions in the classical case. We prove that are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel–Weil theorem and obtain a q-character formula for the characters of irreducible Sp2n-modules via the Atiyah–Bott–Lefschetz fixed points formula.

scholarly journals Degenerate Flag Varieties of Type A and C are Schubert Varieties

2014 ◽  
Vol 2015 (15) ◽  
pp. 6353-6374 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Martina Lanini
Keyword(s):  
Type A ◽  
Schubert Varieties ◽  
Flag Varieties ◽  
Degenerate Flag Varieties

scholarly journals Degenerate flag varieties: moment graphs and Schröder numbers

2012 ◽  
Vol 38 (1) ◽  
pp. 159-189 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Evgeny Feigin ◽  
Markus Reineke
Keyword(s):  
Flag Varieties ◽  
Schröder Numbers ◽  
Degenerate Flag Varieties
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