B-Stable Ideals in the Nilradical of a Borel Subalgebra

By E. Sommers. Canadian mathematical bulletin, 2003

86 citations recorded by the source corpus.

Abstract We count the number of strictly positive $B$ -stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$ -stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/de39bc52d93561774aec9b306f5931ed9f6dbdcd

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Citations in the corpus

Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups (2006, background): “Also, since the elements of an antichain are linearly independent (see [116]), this embedding sends an antichain of cardinality i to a subspace of codimension i.”
Generalized cluster complexes and Coxeter combinatorics (2005, unknown): “These Narayana numbers have been shown [13, 19, 23, 25] to count various combinatorial objects associated with the root system Φ, the Coxeter group W , the corresponding semisimple Lie algebra, etc.”
Root systems and generalized associahedra (2005, background): “ns of N(Φ) given in Section 5.1. These enumerative results are listed in Theorem 5.9 below; we elaborate on the items in the theorem in subsequent comments. Theorem 5.9 is a combination of results in [2, 19, 39, 44, 48]; see [2] for a historical overview, and for further generalizations. LECTURE 5. ”
On a refinement of the generalized Catalan numbers for Weyl groups (2004, background): “1) by Panyushev [14] and is also implicit in the recent work of Sommers [22] (see Proposition 6.”
Abelian ideals in a Borel subalgebra of a complex simple Lie algebra (2002, background): “A natural generalization of Peterson’s approach from abelian to ad-nilpotent ideals was developed recently by several authors [AKOP,CP1,CP2,CP3,KOP, Som ], see also [Shi], and for Kostant’s results [CMP].”
Shellability of noncrossing partition lattices (2005, methodology): “For Coxeter groups of types A and B in the Cartan-Killing classification it is isomorphic to the classical lattice of noncrossing partitions of an n-element set, defined and studied by Kreweras [10], and to its type B analogue, defined by Reiner [14], respectively.”
Affine permutations and rational slope parking functions (2014, unknown): “It has been remarked in [2, 11, 27] that the inverses of the affine permutations lab eling the minimal alcoves in Shi regions belong to a certain simplexD n , which is isometric to the(n + 1)-dilated fundamental alcove.”
Exponents for B-stable ideals (2004, background): “Recommended Citation Sommers, E and Tymoczko, J, "Exponents for B-stable ideals" (2006). TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY.”