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Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers $m_1, m_2, ..., m_k$ which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types A_n, B_n, C_n and some other types. When I is zero, we recover the usual exponents of G by Kostant and one of our conjectures reduces to a well-known factorization of the Poincare polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.
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 compute
We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal
The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalan number of such subspaces. Each ad-nilpo
We define the notion of a power stable ideal in a polynomial ring $ R[X]$ over an integral domain $ R $. It is proved that a maximal ideal $chi$ $ M $ in $ R[X]$ is power stable if and only if $ P^t $ is $ P$- primary for all $ tgeq 1 $ for the prime
Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ whi