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Jacopo Gandini

Publications and source records attributed to Jacopo Gandini.

17 recordsLinked to original sources
arXiv2022

Fully commutative elements and spherical nilpotent orbits

Let g be a simple Lie algebra, with fixed Borel subalgebra b and with Weyl group W. Expanding on previous work of Fan and Stembridge in the simply laced case, this note aims to study the fully commutative elements of W, and their connections with the spherical nilpotent orbits in g. If g is not of type G_2, it is shown that an element w in W is fully commutative if and only if the subalgebra of b determined by the inversions of w lies in the closure of a spherical nilpotent orbit. A similar characterization is also given for the ad-nilpotent ideals of b, which are parametrized by suitable elements in the affine Weyl group of g thanks to the work of Cellini and Papi.

math.RT↗
arXiv2021

On the multiplication of spherical functions of reductive spherical pairs of type A

Let G be a simple complex algebraic group and let K be a reductive subgroup of G such that the coordinate ring of G/K is a multiplicity free G-module. We consider the G-algebra structure of C[G/K], and study the decomposition into irreducible summands of the product of irreducible G-submodules in C[G/K]. When the spherical roots of G/K generate a root system of type A we propose a conjectural decomposition rule, which relies on a conjecture of Stanley on the multiplication of Jack symmetric functions. With the exception of one case, we show that the rule holds true whenever the root system generated by the spherical roots of G/K is direct sum of subsystems of rank one.

math.RT↗
arXiv2021

Some combinatorial properties of skew Jack symmetric functions

Motivated by Stanley's conjecture on the multiplication of Jack symmetric functions, we prove a couple of identities showing that skew Jack symmetric functions are semi-invariant up to translation and rotation of a $π$ angle of the skew diagram. It follows that, in some special cases, the coefficients of the skew Jack symmetric functions with respect to the basis of the monomial symmetric functions are polynomials with nonnegative integer coefficients.

math.CO↗
arXiv2020

Nilpotent orbits of height 2 and involutions in the affine Weyl group

Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B be a Borel subgroup of G. Then B acts with finitely many orbits on the variety N_2 of the nilpotent elements in g whose height is at most 2. We provide a parametrization of the B-orbits in N_2 in terms of subsets of pairwise orthogonal roots, and we provide a complete description of the inclusion order among the B-orbit closures in terms of the Bruhat order on certain involutions in the affine Weyl group of g.

math.AG↗
arXiv2020

The Bruhat order on abelian ideals of Borel subalgebras

Let G be a quasi simple algebraic group over an algebraically closed field k whose characteristic is not very bad for G, and let B be a Borel subgroup of G with Lie algebra b. Given a B-stable abelian subalgebra a of the nilradical of b, we parametrize the B-orbits in a and we describe their closure relations.

math.AG↗
arXiv2018

The Bruhat order on Hermitian symmetric varieties and on abelian nilradicals

Let $G$ be a simple algebraic group and $P$ a parabolic subgroup of $G$ with abelian unipotent radical $P^u$, and let $B$ be a Borel subgroup of $G$ contained in P. Let $\mathfrak{p}^u$ be the Lie algebra of $P^u$ and let $L$ be a Levi factor of $P$, then $L$ is a Hermitian symmetric subgroup of $G$ and $B$ acts with finitely many orbits both on $\mathfrak{p}^u$ and on $G/L$. In this paper we study the Bruhat order of the $B$-orbits in $\mathfrak{p}^u$ and in $G/L$, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.

math.AG↗
arXiv2017

Orbits of strongly solvable spherical subgroups on the flag variety

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup H of B acting with finitely many orbits on the flag variety G/B, and we classify the H-orbits in G/B in terms of suitable combinatorial invariants. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G/B, and we give a combinatorial model for this action in terms of weight polytopes.

math.AG↗
arXiv2016

Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak g$, let $G_0 \subset G$ be the symmetric subgroup defined by an algebraic involution $σ$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy representation of $G_0$. Given an abelian subalgebra $\mathfrak a$ of $\mathfrak g$ contained in $\mathfrak g_1$ and stable under the action of some Borel subgroup $B_0 \subset G_0$, we classify the $B_0$-orbits in $\mathfrak a$ and we characterize the sphericity of $G_0 \mathfrak a$. Our main tool is the combinatorics of $σ$-minuscule elements in the affine Weyl group of $\mathfrak g$ and that of strongly orthogonal roots in Hermitian symmetric spaces.

math.RT↗
arXiv2015

Projective normality of model varieties and related results

We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety M of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and reduces the study of the surjectivity for every couple of globally generated line bundles to a finite number of cases. As a consequence, the cone defined by a complete linear system over M or over a closed G-stable subvariety of M is normal. We apply these results to the study of the normality of the compactifications of model varieties in simple projective spaces and of the closures of the spherical nilpotent orbits. Then we focus on a particular case proving two specific conjectures of Adams, Huang and Vogan on an analogue of the model orbit of the group of type E8.

math.AG↗
arXiv2015

Standard monomial theory for wonderful varieties

A general setting for a standard monomial theory on a multiset is introduced and applied to the Cox ring of a wonderful variety. This gives a degeneration result of the Cox ring to a multicone over a partial flag variety. Further, we deduce that the Cox ring has rational singularities.

math.AG↗
arXiv2012

Normality and smoothness of simple linear group compactifications

If G is a complex semisimple algebraic group, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant GxG-compactifications which possess a unique closed orbit and which arise in a projective space of the shape P(End(V)), where V is finite dimensional rational G-module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of V. In particular, we show that Sp(2r) (with r > 0) is the unique non-adjoint simple group which admits a simple smooth compactification.

math.AG↗
arXiv2012

Simple linear compactifications of odd orthogonal groups

We classify the simple linear compactifications of SO(2r+1), namely those compactifications with a unique closed orbit which are obtained by taking the closure of the SO(2r+1)xSO(2r+1)-orbit of the identity in a projective space P(End(V)), where V is a finite dimensional rational SO(2r+1)-module.

math.AG↗
arXiv2011

Normality and non-normality of group compactifications in simple projective spaces

If $G$ is a complex simply connected semisimple algebraic group and if $λ$ is a dominant weight, we consider the compactification $X_λ$ in the projectivisation of $\End(V(λ))$ obtained as the closure of the $G\times G$-orbit of the identity and we give necessary and sufficient conditions on the support of $λ$ so that $X_λ$ is normal; as well, we give necessary and sufficient conditions on the support of $λ$ so that $X_λ$ is smooth.

math.AG↗
arXiv2010

Spherical orbit closures in simple projective spaces and their normalizations

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module of finite dimension. If G/H \subset P(V) is a spherical orbit and if X is its closure, then we describe the orbits of X and those of its normalization. If moreover the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup.

math.AG↗