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Augustin-Liviu Mare

Publications and source records attributed to Augustin-Liviu Mare.

At least 19 recordsLinked to original sources
arXiv2026

Flag manifolds, spaces of frames, and connectedness properties

For integers $1\le k < n$ and real numbers $c_1, \ldots, c_k>0$ and $d_1, \ldots, d_n\ge 0$, we investigate the space of all $k\times n$ matrices $F$ such that the product $FF^*$ is equal to the diagonal matrix ${\rm Diag}(c_1, \ldots, c_k)$ and the squared norms of the columns of $F$ are equal to $d_1, \ldots, d_n$ respectively. Depending on the field where the coefficients of $F$ are taken from, which can be of real or of complex numbers, we are mainly interested in determining whether the resulting space is path-connected or even simply connected relative to the subspace topology in the space of all $k \times n$ matrices. The criteria presented are closely related to results previously obtained by Cahill, Mixon, and Strawn (2017), Needham and Shonkwiler (2021), the last two authors together with Caine (2026), and the author of this work (2024 and 2026). Flag manifolds, namely orbits of the canonical conjugation actions of ${\rm O}(n)$ and ${\rm U}(n)$ on the spaces of symmetric real and Hermitian $n\times n$ matrices, respectively, play a central role in our development.

math.DG↗
arXiv2026

Simply connectedness of spaces of tight frames

Complex tight frames can be canonically viewed as elements of a complex Stiefel manifold. We present a class of spaces of such frames which are simply connected relative to the subspace topology. To this class belongs the space of finite unit-norm tight frames.

math.FA↗
arXiv2024

Connectivity properties of the Schur-Horn map for real Grassmannians

To any $V$ in the Grassmannian ${\rm Gr}_k({\mathbb R}^n)$ of $k$-dimensional vector subspaces in ${\mathbb R}^n$ one can associate the diagonal entries of the ($n\times n$) matrix corresponding to the orthogonal projection of ${\mathbb R}^n$ to $V$. One obtains a map ${\rm Gr}_k({\mathbb R}^n)\to {\mathbb R}^n$ (the Schur-Horn map). The main result of this paper is a criterion for pre-images of vectors in ${\mathbb R}^n$ to be connected. This will allow us to deduce connectivity criteria for a certain class of subspaces of the real Stiefel manifold which arise naturally in frame theory. We extend in this way results of Cahill, Mixon, and Strawn.

math.DG↗
arXiv2017

Assignments for topological group actions

A polynomial assignment for a continuous action of a compact torus $T$ on a topological space $X$ assigns to each $p\in X$ a polynomial function on the Lie algebra of the isotropy group at $p$ in such a way that a certain compatibility condition is satisfied. The space ${\mathcal{A}}_T(X)$ of all polynomial assignments has a natural structure of an algebra over the polynomial ring of ${\rm Lie}(T)$. It is an equivariant homotopy invariant, canonically related to the equivariant cohomology algebra. In this paper we prove various properties of ${\mathcal{A}}_T(X)$ such as Borel localization, a Chang-Skjelbred lemma, and a Goresky-Kottwitz-MacPherson presentation. In the special case of Hamiltonian torus actions on symplectic manifolds we prove a surjectivity criterion for the assignment equivariant Kirwan map corresponding to a circle in $T$. We then obtain a Tolman-Weitsman type presentation of the kernel of this map.

math.AT↗
arXiv2017

Equivariant cohomology of cohomogeneity one actions: the topological case

We show that for any cohomogeneity one continuous action of a compact connected Lie group $G$ on a closed topological manifold the equivariant cohomology equipped with its canonical $H^*(BG)$-module structure is Cohen-Macaulay. The proof relies on the structure theorem for these actions recently obtained by Galaz-Garcia and Zarei. We generalize in this way our previous result concerning smooth actions.

math.AT↗
arXiv2016

On the complete integrability of the periodic quantum Toda lattice

We prove that the periodic quantum Toda lattice corresponding to any extended Dynkin diagram is completely integrable. This has been conjectured and proved in all classical cases and $E_6$ by Goodman and Wallach at the beginning of the 1980's. As a direct application, in the context of quantum cohomology of affine flag manifolds, results that were known to hold only for some particular Lie types can now be extended to all types.

math.DS↗
arXiv2016

An affine deformation of the quantum cohomology ring of flag manifolds and periodic Toda lattice

Consider the generalized flag manifold $G/B$ and the corresponding affine flag manifold $\mathcal{Fl}_G$. In this paper we use curve neighborhoods for Schubert varieties in $\mathcal{Fl}_G$ to construct certain affine Gromov-Witten invariants of $\mathcal{Fl}_G$, and to obtain a family of "affine quantum Chevalley" operators $Λ_0, \ldots, Λ_n$ indexed by the simple roots in the affine root system of $G$. These operators act on the cohomology ring $\mathrm{H}^*(\mathcal{Fl}_G)$ with coefficients in $\mathbb{Z}[q_0, \ldots,q_n]$. By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for $G= \mathrm{SL}_n(\mathbb{C})$. The first quantum ring is a deformation of the subalgebra of $\mathrm{H}^*(\mathcal{Fl}_G)$ generated by divisors. The second ring, denoted $\mathrm{QH}^*_{\mathrm{af}}(G/B)$, deforms the ordinary quantum cohomology ring $\mathrm{QH}^*(G/B)$ by adding an affine quantum parameter $q_0$. We prove that $\mathrm{QH}^*_{\mathrm{af}}(G/B)$ is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of $\mathrm{QH}^*_{\mathrm{af}}(G/B)$ by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of $G$.

math.AG↗
arXiv2013

Topology of the octonionic flag manifold

The octonionic flag manifold $Fl(\mathbb{O})$ is the space of all pairs in $\mathbb{O}P^2\times \mathbb{O}P^2$ (where $\mathbb{O}P^2$ denotes the octonionic projective plane) which satisfy a certain "incidence" relation. It comes equipped with the projections $π_1,π_2 : Fl(\mathbb{O})\to \mathbb{O}P^2$, which are $\mathbb{O}P^1$ bundles, as well as with an action of the group $Spin(8)$. The first two results of this paper give Borel type descriptions of the usual, respectively $Spin(8)$-equivariant cohomology of $Fl(\mathbb{O})$ in terms of $π_1$ and $π_2$ (actually the Euler classes of the tangent spaces to the fibers of $π_1$, respectively $π_2$, which are rank 8 vector bundles on $Fl(\mathbb{O})$). Then we obtain a Goresky-Kottwitz-MacPherson type description of the ring $H^*_{Spin(8)}(Fl(\mathbb{O}))$. Finally, we consider the $Spin(8)$-equivariant $K$-theory ring of $Fl(\mathbb{O})$ and obtain a Goresky-Kottwitz-MacPherson type description of this ring.

math.AT↗
arXiv2012

Non-abelian GKM Theory

We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting group. We show that the algebra structure of the equivariant cohomology can be read off from this graph. In comparison with ordinary abelian GKM theory, there are some special features due to the more complicated structure of the non-abelian 1-skeleton.

math.DG↗
arXiv2012

Bott periodicity for inclusions of symmetric spaces

When looking at Bott's original proof of his periodicity theorem for the stable homotopy groups of the orthogonal and unitary groups, one sees in the background a differential geometric periodicity phenomenon. We show that this geometric phenomenon extends to the standard inclusion of the orthogonal group into the unitary group. Standard inclusions between other classical Riemannian symmetric spaces are considered as well. An application to homotopy theory is also discussed.

math.DG↗
arXiv2011

Equivariant cohomology of cohomogeneity one actions

We show that if $G\times M \to M$ is a cohomogeneity one action of a compact connected Lie group $G$ on a compact connected manifold $M$ then $H^*_G(M)$ is a Cohen-Macaulay module over $H^*(BG)$. Moreover, this module is free if and only if the rank of at least one isotropy group is equal to the rank of $G$. We deduce as corollaries several results concerning the usual (de Rham) cohomology of $M$, such as the following obstruction to the existence of a cohomogeneity one action: if $M$ admits a cohomogeneity one action, then $χ(M)>0$ if and only if $H^{\rm odd}(M)=\{0\}$.

math.DG↗
arXiv2010

A quantum type deformation of the cohomology ring of flag manifolds

Let q_1, ..., q_n be some variables and set K:=Z[q_1, ..., q_n]/(q_1q_2...q_n). We show that there exists a K-bilinear product \star on H^*(F_n;Z)\otimes K which is uniquely determined by some quantum cohomology like properties (most importantly, a degree two relation involving the generators and an analogue of the flatness of the Dubrovin connection). Then we prove that \star satisfies the Frobenius property with respect to the Poincaré pairing of H^*(F_n;Z); this leads immediately to the orthogonality of the corresponding Schubert type polynomials. We also note that if we pick k\in {1,...,n} and we formally replace q_k by 0, the ring (H^*(F_n;Z)\otimes K,\star) becomes isomorphic to the usual small quantum cohomology ring of F_n, by an isomorphism which is described precisely.

math.CO↗
arXiv2009

Real loci of based loop groups

Let $(G,K)$ be a Riemannian symmetric pair of maximal rank, where $G$ is a compact simply connected Lie group and $K$ the fixed point set of an involutive automorphism $σ$. This induces an involutive automorphism $τ$ of the based loop space $Ω(G)$. There exists a maximal torus $T\subset G$ such that the canonical action of $T\times S^1$ on $Ω(G)$ is compatible with $τ$ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat's convexity theorem. Namely, the images of $Ω(G)$ and $Ω(G)^τ$ (fixed point set of $τ$) under the $T\times S^1$ moment map on $Ω(G)$ are equal. The space $Ω(G)^τ$ is homotopy equivalent to the loop space $Ω(G/K)$ of the Riemannian symmetric space $G/K$. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in $\mathbb{Z}_2$ of $Ω(G)$ and $Ω(G/K)$. Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved. The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe.

math.DG↗
arXiv2009

Equivariant $K$-theory of quaternionic flag manifolds

We consider the manifold $Fl_n(\mathbb{H})=Sp(n)/Sp(1)^n$ of all complete flags in $\mathbb{H}^n$, where $\mathbb{H}$ is the skew-field of quaternions. We study its equivariant $K$-theory rings with respect to the action of two groups: $Sp(1)^n$ and a certain canonical subgroup $T:=(S^1)^n\subset Sp(1)^n$ (a maximal torus). For the first group action we obtain a Goresky-Kottwitz-MacPherson type description. For the second one, we describe the ring $K_T(Fl_n(\mathbb{H}))$ as a subring of $K_T(Sp(n)/T)$. This ring is well known, since $Sp(n)/T$ is a complex flag variety.

math.AT↗
arXiv2009

On some symplectic quotients of Schubert varieties

Let $G/P$ be a generalized flag variety, where $G$ is a complex semisimple connected Lie group and $P\subset G$ a parabolic subgroup. Let also $X\subset G/P$ be a Schubert variety. We consider the canonical embedding of $X$ into a projective space, which is obtained by identifying $G/P$ with a coadjoint orbit of the compact Lie group $K$, where $G=K^{\mathbb C}$. The maximal torus $T$ of $K$ acts linearly on the projective space and it leaves $X$ invariant: let $Ψ: X \to {\rm Lie}(T)^*$ be the restriction of the moment map relative to the Fubini-Study symplectic form. By a theorem of Atiyah, $Ψ(X)$ is a convex polytope in ${\rm Lie}(T)^*$. In this paper we show that all pre-images $Ψ^{-1}(μ)$, $μ\in Ψ(X)$, are connected subspaces of $X$. We then consider a one-dimensional subtorus $S\subset T$, and the map $f: X\to {\mathbb R}$, which is the restriction of the $S$ moment map to $X$. We study quotients of the form $f^{-1}(r)/S$, where $r\in {\mathbb R}$. We show that under certain assumptions concerning $X$, $S$, and $r$, these symplectic quotients are (new) examples of spaces for which the Kirwan surjectivity theorem and Tolman and Weitsman's presentation of the kernel of the Kirwan map hold true (combined with a theorem of Goresky, Kottwitz, and MacPherson, these results lead to an explicit description of the cohomology ring of the quotient). The singular Schubert variety in the Grassmannian $G_2({\mathbb C}^4)$ of 2 planes in ${\mathbb C}^4$ is discussed in detail.

math.SG↗
arXiv2009

Connectivity properties of moment maps on based loop groups

For a compact, connected, simply-connected Lie group G, the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H^1-Sobolev maps S^1-->G. The geometry of LG and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops Omega(G) is an example of a homogeneous space of $LG$ and has a natural Hamiltonian T x S^1 action, where T is the maximal torus of G. We study the moment map mu for this action, and in particular prove that its regular level sets are connected. This result is as an infinite-dimensional analogue of a theorem of Atiyah that states that the preimage of a moment map for a Hamiltonian torus action on a compact symplectic manifold is connected. In the finite-dimensional case, this connectivity result is used to prove that the image of the moment map for a compact Hamiltonian T-space is convex. Thus our theorem can also be viewed as a companion result to a theorem of Atiyah and Pressley, which states that the image mu(Omega(G)) is convex. We also show that for the energy functional E, which is the moment map for the S^1 rotation action, each non-empty preimage is connected.

math.SG↗