ترغب بنشر مسار تعليمي؟ اضغط هنا

B-stable ideals in the nilradical of a Borel subalgebra

83   0   0.0 ( 0 )
 نشر من قبل Eric Sommers
 تاريخ النشر 2003
  مجال البحث
والبحث باللغة English
 تأليف Eric Sommers




اسأل ChatGPT حول البحث

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 the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.



قيم البحث

اقرأ أيضاً

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 propos e 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.
61 - Molly Fenn , Eric Sommers 2021
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 tent ideal meets a unique largest nilpotent orbit in the Lie algebra of all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studied by Mizuno and others, the equivalence classes are the ad-nilpotent ideals with the same largest nilpotent orbit. We include two applications of the result, one to the higher vanishing of cohomology groups of vector bundles on the flag variety and another to the Kazhdan-Lusztig cells in the affine Weyl group of the symmetric group. Finally, some combinatorial results are discussed.
Fix a poset $Q$ on ${x_1,ldots,x_n}$. A $Q$-Borel monomial ideal $I subseteq mathbb{K}[x_1,ldots,x_n]$ is a monomial ideal whose monomials are closed under the Borel-like moves induced by $Q$. A monomial ideal $I$ is a principal $Q$-Borel ideal, deno ted $I=Q(m)$, if there is a monomial $m$ such that all the minimal generators of $I$ can be obtained via $Q$-Borel moves from $m$. In this paper we study powers of principal $Q$-Borel ideals. Among our results, we show that all powers of $Q(m)$ agree with their symbolic powers, and that the ideal $Q(m)$ satisfies the persistence property for associated primes. We also compute the analytic spread of $Q(m)$ in terms of the poset $Q$.
Let $dge1$ be an integer, $W_d$ and $mathcal{K}_d$ be the Witt algebra and the weyl algebra over the Laurent polynomial algebra $A_d=mathbb{C} [x_1^{pm1}, x_2^{pm1}, ..., x_d^{pm1}]$, respectively. For any $mathfrak{gl}_d$-module $M$ and any admissib le module $P$ over the extended Witt algebra $widetilde W_d$, we define a $W_d$-module structure on the tensor product $Potimes M$. We prove in this paper that any simple $W_d$-module that is finitely generated over the cartan subalgebra is a quotient module of the $W_d$-module $P otimes M$ for a finite dimensional simple $mathfrak{gl}_d$-module $M$ and a simple $mathcal{K}_d$-module $P$ that are finitely generated over the cartan subalgebra. We also characterize all simple $mathcal{K}_d$-modules and all simple admissible $widetilde W_d$-modules that are finitely generated over the cartan subalgebra.
An equigenerated monomial ideal $I$ is a Freiman ideal if $mu(I^2)=ell(I)mu(I)-{ell(I)choose 2}$ where $ell(I)$ is the analytic spread of $I$ and $mu(I)$ is the least number of monomial generators of $I$. Freiman ideals are special since there exists an exact formula computing the least number of monomial generators of any of their powers. In this paper we give a complete classification of Freiman $t$-spread principal Borel ideals.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا