degenerate flag varieties
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2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ghislain Fourier ◽  
Gabriele Nebe

<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>



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


2014 ◽  
Vol 66 (6) ◽  
pp. 1250-1286 ◽  
Author(s):  
Evgeny Feigin ◽  
Michael Finkelberg ◽  
Peter Littelmann

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.



2014 ◽  
Vol 2015 (15) ◽  
pp. 6353-6374 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Martina Lanini


10.37236/4068 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Ange Bigeni

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 $\bar{c}_n(q)$.



2012 ◽  
Vol 275 (1-2) ◽  
pp. 55-77 ◽  
Author(s):  
Evgeny Feigin ◽  
Michael Finkelberg


2012 ◽  
Vol 38 (1) ◽  
pp. 159-189 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Evgeny Feigin ◽  
Markus Reineke


2012 ◽  
Vol 6 (1) ◽  
pp. 165-194 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Evgeny Feigin ◽  
Markus Reineke




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