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.
Refereed journal articles
Mare, A.-L. (2024). Connectivity properties of the Schur-Horn map for real Grassmannians. Abh. Math. Semin. Univ. Hamburg, 94, 33–55.
@article{feynman1939forces,
title = {Connectivity properties of the Schur-Horn map for real Grassmannians},
author = {Mare, A.-L.},
journal = {Abh. Math. Semin. Univ. Hamburg},
volume = {94},
%number = {4},
pages = {33 - 55},
year = {2024},
%note = {2309.11596},
eprint = {2309.11596},
archiveprefix = {arXiv},
primaryclass = {math},
doi = {10.1007/s12188-024-00277-1},
%file = {feynman39.pdf}
}
To any V in the Grassmannian Gr_k(ℝ^n) of k-dimensional real vector subspaces in ℝ^n one can associate the diagonal entries of the (n×n) matrix corresponding to the orthogonal projection of R^n to V. One obtains a map Gr_k(ℝ^n) → ℝ^n (the Schur-Horn map). The main result of this paper is a criterion for pre-images of vectors in 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.
Carlson, J. D., Goertsches, O., He, C., & Mare, A.-L. (2019). The equivariant cohomology ring of a cohomogeneity-one action. Geom. Dedicata, 203, 205–223.
@article{feynman1939forcet,
title = {The equivariant cohomology ring of a cohomogeneity-one action},
author = {Carlson, J. D. and Goertsches, O. and He, C. and Mare, A.-L.},
journal = {Geom. Dedicata},
volume = {203},
%number = {4},
pages = {205 - 223},
year = {2019},
eprint = {1802.02304},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1007/s10711-019-00434-4},
%file = {feynman39.pdf}
}
We compute the rational Borel equivariant cohomology ring of a cohomogeneity-one action of a compact Lie group.
Goertsches, O., Noshari, S. H. S., & Mare, A.-L. (2019). On the equivariant cohomology of hyperpolar actions on symmetric spaces. Doc. Math., 24, 205–223.
@article{feynman1939forceu,
title = {On the equivariant cohomology of hyperpolar actions on symmetric spaces},
author = {Goertsches, O. and Noshari, S. H. S. and Mare, A.-L.},
journal = {Doc. Math.},
volume = {24},
%number = {4},
pages = {205 - 223},
year = {2019},
eprint = {1808.10630},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.4171/DM/711},
%file = {feynman39.pdf}
}
We show that the equivariant cohomology of any hyperpolar action of a compact and connected Lie group on a symmetric space of compact type is a Cohen-Macaulay ring. This generalizes some results previously obtained by the authors.
Mare, A.-L., & Mihalcea, L. C. (2017). An affine deformation of the quantum cohomology ring of flag manifolds and periodic Toda lattice. Proc. Lond. Math. Soc., 116(4), 135–181.
@article{feynman1939forcev,
title = {An affine deformation of the quantum cohomology ring of flag manifolds and periodic Toda lattice},
author = {Mare, A.-L. and Mihalcea, L. C.},
journal = {Proc. Lond. Math. Soc.},
volume = {116},
number = {4},
pages = {135 - 181},
year = {2017},
eprint = {1409.3587},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1112/plms.12077},
%file = {feynman39.pdf}
}
Consider the generalized flag manifold G/B and the corresponding affine flag manifold Fl_G. In this paper we use curve neighborhoods for Schubert varieties in
Fl_G to construct certain affine Gromov-Witten invariants of Fl_G, and to obtain a family of "affine quantum Chevalley" operators \Lambda_0, ... , \Lambda_n indexed by the simple roots in the affine root system of G. These operators act on the cohomology ring H^*(Fl_G) with coefficients in ℤ[q_0, ..., 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=SL_n(ℂ). The first quantum ring is a deformation of the subalgebra of H^*(Fl_G) generated by divisors. The second ring, denoted QH^*_af(G/B), deforms the ordinary quantum cohomology ring QH^*(G/B) by adding an affine quantum parameter q_0. We prove that QH^*_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 QH^*_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.
Mare, A.-L. (2017). On the complete integrability of the periodic quantum Toda lattice. Forum Math., 29(6), 1413–1428.
@article{feynman1939forcew,
title = {On the complete integrability of the periodic quantum Toda lattice},
author = {Mare, A.-L.},
journal = {Forum Math.},
volume = {29},
number = {6},
pages = {1413 - 1428},
year = {2017},
eprint = {1604.08726},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1515/forum-2016-0117},
%file = {feynman39.pdf}
}
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 E6 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.
Goertsches, O., & Mare, A.-L. (2017). Assignments for topological group actions. Indag. Math., 28(6), 1210–1232.
@article{feynman1939forcex,
title = {Assignments for topological group actions},
author = {Goertsches, O. and Mare, A.-L.},
journal = {Indag. Math.},
volume = {28},
number = {6},
pages = {1210 - 1232},
year = {2017},
eprint = {1512.06579 },
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1016/j.indag.2017.09.005},
%file = {feynman39.pdf}
}
A polynomial assignment for a continuous action of a compact torus T on a topological space X assigns to each p∈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 A_T(X) of all polynomial assignments has a natural structure of an algebra over the polynomial ring of Lie(T). It is an equivariant homotopy invariant, canonically related to the equivariant cohomology algebra. In this paper we prove various properties of 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.
Goertsches, O., & Mare, A.-L. (2017). Equivariant cohomology of cohomogeneity one actions: the topological case. Topology Appl., 218(6), 93–96.
@article{feynman1939forcey,
title = {Equivariant cohomology of cohomogeneity one actions: the topological case},
author = {Goertsches, O. and Mare, A.-L.},
journal = {Topology Appl.},
volume = {218},
number = {6},
pages = {93 - 96},
year = {2017},
eprint = {1609.07316 },
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1016/j.topol.2016.12.017},
%file = {feynman39.pdf}
}
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.
@article{feynman1939forcez,
title = {Non-abelian GKM theory},
author = {Goertsches, O. and Mare, A.-L.},
journal = {Math. Z.},
volume = {277},
%number = {6},
pages = {1 - 27},
year = {2014},
eprint = {1208.5568},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1007/s00209-013-1242-x},
%file = {feynman39.pdf}
}
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.
Goertsches, O., & Mare, A.-L. (2014). Equivariant cohomology of cohomogeneity one actions. Topology Appl., 167, 36–52.
@article{feynman1939forcfa,
title = {Equivariant cohomology of cohomogeneity one actions},
author = {Goertsches, O. and Mare, A.-L.},
journal = {Topology Appl.},
volume = {167},
%number = {6},
pages = {36 - 52},
year = {2014},
eprint = {1110.6318},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1016/j.topol.2014.03.006},
%file = {feynman39.pdf}
}
We show that if G x M→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 rank G. We deduce as corollaries several results concerning the usual (de Rham) cohomology of M, such as a new proof of 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^odd(M) = 0.
Mare, A.-L., & Willems, M. (2013). Topology of the octonionic flag manifold. Münster J. of Math., 6, 483–523.
@article{feynman1939forcfb,
title = {Topology of the octonionic flag manifold},
author = {Mare, A.-L. and Willems, M.},
journal = {M\"unster J. of Math.},
volume = {6},
%number = {6},
pages = {483 - 523},
year = {2013},
eprint = {0809.4318 },
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.48550/arXiv.0809.4318},
%file = {feynman39.pdf}
}
The octonionic flag manifold Fl(𝕆) is the space of all pairs in 𝕆P^2×𝕆P^2 (where 𝕆P^2 denotes the octonionic projective plane) which satisfy a certain "incidence" relation. It comes equipped with the projections π1,π2:Fl(𝕆)→𝕆P^2, which are 𝕆P^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(𝕆) in terms of π1 and π2 (actually the Euler classes of the tangent spaces to the fibers of π1, respectively π2, which are rank 8 vector bundles on Fl(𝕆)). Then we obtain a Goresky-Kottwitz-MacPherson type description of the ring H^*_Spin(8)(Fl(𝕆)). Finally, we consider the Spin(8)-equivariant K-theory ring of Fl(𝕆) and obtain a Goresky-Kottwitz-MacPherson type description of this ring.
Mare, A.-L., & Quast, P. (2012). Bott periodicity for inclusions of symmetric spaces. Doc. Math., 17, 911–952.
@article{feynman1939forcfc,
title = {Bott periodicity for inclusions of symmetric spaces},
author = {Mare, A.-L. and Quast, P.},
journal = {Doc. Math.},
volume = {17},
%number = {6},
pages = {911 - 952},
year = {2012},
eprint = {1108.0954},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.4171/DM/385},
%file = {feynman39.pdf}
}
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.
Mare, A.-L., & Quast, P. (2012). On some spaces of minimal geodesics in Riemannian symmetric spaces. Quart. J. Math., 63(3), 681–694.
@article{feynman1939forcfd,
title = {On some spaces of minimal geodesics in Riemannian symmetric spaces},
author = {Mare, A.-L. and Quast, P.},
journal = {Quart. J. Math.},
volume = {63},
number = {3},
pages = {681 - 694},
year = {2012},
%eprint = {1108.0954},
%archiveprefix = {arXiv},
%primaryclass = {math},
%note = {1802.02304},
doi = {10.1093/qmath/har003},
%file = {feynman39.pdf}
}
Let (P, o) be a pointed irreducible symmetric space of compact type. In this paper, we study the space of minimal (i.e. shortest) geodesics in P from o to a point in the center of (P, o). More precisely, we first give a complete description of these geodesics. We then show that each connected component of the locus described by the midpoints of these geodesics is an extrinsically symmetric space (symmetric R-space) which is totally geodesically and isometrically embedded in P.
Eschenburg, J.-H., Mare, A.-L., & Quast, P. (2010). Pluriharmonic maps into outer symmetric spaces and a subdivison of Weyl chambers. Bull. London Math. Soc., 42(6), 1121–1133.
@article{feynman1939forcfe,
title = {Pluriharmonic maps into outer symmetric spaces and a subdivison of Weyl chambers},
author = {Eschenburg, J.-H. and Mare, A.-L. and Quast, P.},
journal = {Bull. London Math. Soc.},
volume = {42},
number = {6},
pages = {1121 - 1133},
year = {2010},
%eprint = {1108.0954},
%archiveprefix = {arXiv},
%primaryclass = {math},
%note = {1802.02304},
doi = {10.1112/blms/bdq070},
%file = {feynman39.pdf}
}
Burstall and Guest have given a classification of harmonic maps of the 2-sphere with values in Lie groups and inner symmetric spaces. We extend their result to outer symmetric spaces G/K, using the pointed Cartan embedding into G. We show that in this case the number of classes can be reduced from 2r to 2s where r = rank G and s = rank K. Moreover we replace the 2-sphere by a simply connected compact Kähler manifold and ‘harmonic’ by ‘pluriharmonic’.
Mare, A.-L. (2010). Connectedness of levels for moment maps on various classes of loop groups. Osaka J. Math., 47(3), 609–626.
@article{feynman1939forcff,
title = {Connectedness of levels for moment maps on various classes of loop groups},
author = {Mare, A.-L.},
journal = {Osaka J. Math.},
volume = {47},
number = {3},
pages = {609 - 626},
year = {2010},
eprint = {math/0702792},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1112/blms/bdq070},
%file = {feynman39.pdf}
}
The space Ω(G) of all based loops in a compact simply connected Lie group G
has an action of the maximal torus T⊂G (by pointwise conjugation) and of the circle
S^1 (by rotation of loops). Let μ: Ω(G)→(t × iℝ)^* be a moment map of the resulting
T × S^1 action. We show that all levels (that is, pre-images of points) of
μ are connected subspaces of Ω(G) (or empty). The result holds if in the definition of Ω(G) loops are of class C^∞ or of any Sobolev class H^s, with s ≥ 1 (for loops of class H^1 connectedness of regular levels has been previously proved by Harada, Holm, Jeffrey, and the author.
Mare, A.-L. (2010). A quantum type deformation of the cohomology ring of flag manifolds. J. Lie Theory, 20, 329–346.
@article{feynman1939forcfg,
title = {A quantum type deformation of the cohomology ring of flag manifolds},
author = {Mare, A.-L.},
journal = {J. Lie Theory},
volume = {20},
%number = {3},
pages = {329 - 346},
year = {2010},
eprint = {math/0610298},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
%doi = {10.1112/blms/bdq070},
file = {marela2e.pdf}
}
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 ∗ on H^*(F_n;ℤ) ⊗ 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 ∗ 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 ∈ {1,...,n} and we formally replace q_k by 0, the ring (H^*(F_n;ℤ) ⊗ K, ∗) becomes isomorphic to the usual small quantum cohomology ring of F_n, by an isomorphism which is described precisely.
Jeffrey, L. C., & Mare, A.-L. (2010). Real loci of based loop groups. Transform. Groups, 15, 134–153.
@article{feynman1939forcfh,
title = {Real loci of based loop groups},
author = {Jeffrey, L. C. and Mare, A.-L.},
journal = {Transform. Groups},
volume = {15},
%number = {3},
pages = {134 - 153},
year = {2010},
eprint = {0903.0840},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1007/S00031-010-9075-8},
%file = {marela2e.pdf}
}
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⊂G such that the canonical action of T×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×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 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.
Mare, A.-L. (2010). On some symplectic quotients of Schubert varieties. Beitr. Algebra Geom., 51(1), 9–30.
Let G/P be a generalized flag variety, where G is a complex semisimple connected Lie group and P⊂G a parabolic subgroup. Let also X⊂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^ℂ. The maximal torus T of K acts linearly on the projective space and it leaves X invariant: let Ψ:X→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 Lie(T)^∗. In this paper we show that all pre-images Ψ^−1(μ), μ∈Ψ(X), are connected subspaces of X. We then consider a one-dimensional subtorus S⊂T, and the map f:X→ℝ, which is the restriction of the S moment map to X. We study quotients of the form f^−1(r)/S, where r∈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(ℂ^4) of 2-planes in ℂ^4 is discussed in detail.
Mare, A.-L., & Willems, M. (2009). Equivariant K-theory of quaternionic flag manifolds. J. K-Theory, 4(3), 537–557.
@article{feynman1939forcfj,
title = {Equivariant K-theory of quaternionic flag manifolds},
author = {Mare, A.-L. and Willems, M.},
journal = {J. K-Theory},
volume = {4},
number = {3},
pages = {537 - 557},
year = {2009},
eprint = {0710.3766},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1017/is009009029jkt082},
%file = {marela2e.pdf}
}
We consider the manifold Fl_n(ℍ)=Sp(n)/Sp(1)^n of all complete flags in ℍ^n, where 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⊂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(ℍ)) as a subring of K_T(Sp(n)/T). The latter ring is well known, since Sp(n)/T is a complex flag variety.
Mare, A.-L. (2008). A characterization of the quantum cohomology ring of G/B and applications. Canad. J. Math., 60, 875–891.
@article{feynman1939forcfk,
title = {A characterization of the quantum cohomology ring of G/B and applications},
author = {Mare, A.-L.},
journal = {Canad. J. Math.},
volume = {60},
%number = {3},
pages = {875 - 891},
year = {2008},
eprint = {math/0311320},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.4153/CJM-2008-037-8},
%file = {marela2e.pdf}
}
We show that the small quantum product of the generalized flag manifold G/B is a product operation on H^*(G/B)⊗ℝ[q_1 ,...,q_l] uniquely determined by the fact that it is a deformation of the cup product on H^*(G/B), it is commutative, associative, graded with respect to deg(q_i) = 4, it satisfies a certain relation (of degree two), and the corresponding Dubrovin connection is flat. We deduce that it is again the flatness of the Dubrovin connection which characterizes essentially the solutions of the “quantum Giambelli problem” for G/B. This result gives new proofs of the quantum Chevalley formula (previously proved by D. Peterson), and of Fomin, Gelfand and Postnikov’s
description of the quantization map for Fl_n(ℂ).
Mare, A.-L. (2008). Equivariant cohomology of quaternionic flag manifolds. J. Algebra, 319(7), 2830–2844.
@article{feynman1939forcfl,
title = {Equivariant cohomology of quaternionic flag manifolds},
author = {Mare, A.-L.},
journal = {J. Algebra},
volume = {319},
number = {7},
pages = {2830 - 2844},
year = {2008},
eprint = {math/0605539},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1016/j.jalgebra.2007.04.031},
%file = {marela2e.pdf}
}
The main result of the paper is a Borel type description of the Sp(1)^n-equivariant cohomology ring of the manifold Fln(ℍ) of all complete flags in ℍ^n. To prove this, we obtain a Goresky-Kottwitz-MacPherson type description of that ring.
Harada, M., Holm, T., Jeffrey, L. C., & Mare, A.-L. (2006). Connectivity properties of moment maps on based loop groups. Geometry and Topology, 10, 1607–1634.
@article{feynman1939forcfm,
title = {Connectivity properties of moment maps on based loop groups},
author = {Harada, M. and Holm, T. and Jeffrey, L. C. and Mare, A.-L.},
journal = {Geometry and Topology},
volume = {10},
%number = {7},
pages = {1607 - 1634},
year = {2006},
eprint = {math/0503684},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.2140/gt.2006.10.1607},
%file = {marela2e.pdf}
}
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 ΩG 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 µ 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 µ(Ω(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.
Eschenburg, J.-H., & Mare, A.-L. (2006). Steepest descent on real flag manifolds. Bull. London Math. Soc., 38(2), 323–328.
@article{feynman1939forcfn,
title = {Steepest descent on real flag manifolds},
author = {Eschenburg, J.-H. and Mare, A.-L.},
journal = {Bull. London Math. Soc.},
volume = {38},
number = {2},
pages = {323 - 328},
year = {2006},
%eprint = {1108.0954},
%archiveprefix = {arXiv},
%primaryclass = {math},
%note = {1802.02304},
doi = {10.1112/S0024609306018376},
%file = {feynman39.pdf}
}
Real flag manifolds are the isotropy orbits of noncompact symmetric spaces G/K. Any such manifold M is acted on transitively by the (noncompact) Lie group G, and it is embedded in euclidean space as a taut submanifold. The aim of this paper is to show that the gradient flow of any height function is a one-parameter subgroup of G, where the gradient is defined with respect to a suitable homogeneous metric s on M; this generalizes the Kähler metric on adjoint orbits (the so-called complex flag manifolds).
Mare, A.-L. (2006). Equivariant cohomology of real flag manifolds. Diff. Geom. Appl., 24(3), 223–229.
@article{feynman1939forcfo,
title = {Equivariant cohomology of real flag manifolds},
author = {Mare, A.-L.},
journal = {Diff. Geom. Appl.},
volume = {24},
number = {3},
pages = {223 - 229},
year = {2006},
eprint = {math/0404369},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1016/j.difgeo.2005.09.006},
%file = {marela2e.pdf}
}
Let P= G/K be a semisimple non-compact Riemannian symmetric space,
where G= I_o(P ) and K= G_p is the stabilizer of p ∈ P . Let X be an orbit of the (isotropy) representation of K on T_p(P ) (X is called a real flag manifold). Let K_0 ⊂ K be the stabilizer of a maximal flat, totally geodesic submanifold of P which contains p. We show that if all the simple root multiplicities of G/K are at least 2 then K_0 is connected and the action of K_0 on X is equivariantly formal. In the case when the multiplicities are equal and at least 2, we will give a purely geometric proof of a formula of Hsiang, Palais and Terng concerning H^*(X). In particular, this gives a conceptually new proof of Borel’s formula for the cohomology ring of an adjoint orbit of a compact Lie group.
Jeffrey, L. C., Mare, A.-L., & Woolf, J. (2005). The Kirwan map, equivariant Kirwan maps and their kernels. J. Reine Angew. Math., 589(3), 105–127.
@article{feynman1939forcfp,
title = {The Kirwan map, equivariant Kirwan maps and their kernels},
author = {Jeffrey, L. C. and Mare, A.-L. and Woolf, J.},
journal = {J. Reine Angew. Math.},
volume = {589},
number = {3},
pages = {105 - 127},
year = {2005},
eprint = {math/0211297},
archiveprefix = {arXiv},
primaryclass = {math},
%note = {1802.02304},
doi = {10.1515/crll.2005.2005.589.105},
%file = {marela2e.pdf}
}
Consider a Hamiltonian action of a compact Lie group K on a compact symplectic manifold. We find descriptions of the kernel of the Kirwan map corresponding to a
regular value of the moment map κ_K. We start with the case when K is a torus T : we determine the kernel of the equivariant Kirwan map (defined by Goldin) corresponding
to a generic circle S⊂T , and show how to recover from this the kernel of κ_T, as described by Tolman and Weitsman. (In the situation when the fixed point set of the torus action is finite, similar results have been obtained in two of our previous papers). For a compact nonabelian Lie group K we will use the “non-abelian localization formula” of Jeffrey and Kirwan to establish relationships — some of them obtained by Tolman and Weitsman — between Ker(κ_K) and Ker(κ_T ), where T ⊂ K is a maximal torus. In the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of µ_T.
Mare, A.-L. (2005). The combinatorial quantum cohomology ring of G/B. Jour. Alg. Comb., 21, 331–349.
@article{mare02,
title = {The combinatorial quantum cohomology ring of G/B},
author = {Mare, A.-L.},
journal = {Jour. Alg. Comb.},
volume = {21},
%number = {3},
pages = {331 - 349},
year = {2005},
eprint = {math/0301257},
archiveprefix = {arXiv},
primaryclass = {math},
%file = {mare02.pdf},
link = {https://link.springer.com/article/10.1007/s10801-005-6915-z}
}
A purely combinatorial construction of the quantum cohomology ring of the flag manifold G/B is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of two of our previous papers, we obtain a presentation of this ring in terms of generators and relations, as well as formulas for quantum Giambelli polynomials. We show that these polynomials satisfy a certain orthogonality property, which - for G=SL_n(ℂ) - was proved previously by Fomin, Gelfand and Postnikov.
Jeffrey, L. C., & Mare, A.-L. (2005). Products of conjugacy classes in SU(2). Canad. Math. Bull., 48(1), 90–96.
@article{mare03,
title = {Products of conjugacy classes in SU(2)},
author = {Jeffrey, L. C. and Mare, A.-L.},
journal = {Canad. Math. Bull.},
volume = {48},
number = {1},
pages = {90 - 96},
year = {2005},
eprint = {math/0211424},
archiveprefix = {arXiv},
primaryclass = {math},
doi = {10.4153/CMB-2005-008-8}
}
We obtain a complete description of collections of n conjugacy classes in SU(2) with the property that the multiplication map from the product of these n conjugacy classes to SU(2) is surjective. The basic instrument is a characterization of collections of n+1 conjugacy classes in SU(2) such that the image of the multiplication map from these conjugacy classes to SU(2) contains the identity matrix, which generalizes a result of Jeffrey and Weitsman.
Mare, A.-L. (2005). Connectivity and Kirwan surjectivity for isoparametric submanifolds. Int. Math. Res. Not., 55, 3427–3443.
@article{mare04,
title = {Connectivity and Kirwan surjectivity for isoparametric submanifolds},
author = {Mare, A.-L.},
journal = {Int. Math. Res. Not.},
volume = {55},
%number = {1},
pages = {3427 - 3443},
year = {2005},
eprint = {math/0505118v1},
archiveprefix = {arXiv},
primaryclass = {math},
doi = {10.1155/IMRN.2005.3427}
}
Atiyah’s formulation of what is nowadays called the convexity theorem of Atiyah-Guillemin-Sternberg has two parts: (a) the image of the moment map arising from a Hamiltonian action of a torus on a symplectic manifold is a convex polytope, and (b) all preimages of the moment map are connected. Part (a) was generalized by Terng to the wider context of isoparametric submanifolds in euclidean space. In this paper we prove a generalization of part (b) for a certain class of isoparametric submanifolds (more precisely, for those with all multiplicities strictly greater than 1). For generalized real flag manifolds, which are an important class of isoparametric submanifolds, we give a surjectivity criterium of a certain Kirwan map (involving equivariant cohomology with rational coefficients) which arises naturally in this context. Examples are also discussed.
Goldin, R. F., & Mare, A.-L. (2004). Cohomology of reduced spaces of generic coadjoint orbits. Proc. Amer. Math. Soc., 132(10), 3069–3074.
@article{mare05,
title = { Cohomology of reduced spaces of generic coadjoint orbits},
author = {Goldin, R. F. and Mare, A.-L.},
journal = {Proc. Amer. Math. Soc.},
volume = {132},
number = {10},
pages = {3069 - 3074},
year = {2004},
eprint = {math/0210434v1},
archiveprefix = {arXiv},
primaryclass = {math},
doi = {10.1090/S0002-9939-04-07443-X}
}
Let O_λ be a generic coadjoint orbit of a compact semi-simple Lie
group K. Weight varieties are the symplectic reductions of O_λ by the maximal
torus T in K. We use a theorem of Tolman and Weitsman to compute the
cohomology ring of these varieties. Our formula relies on a Schubert basis of
the equivariant cohomology of O_λ, and it makes explicit the dependence on λ
and a parameter in Lie(T )^∗.
Mare, A.-L. (2004). Quantum cohomology of the infinite dimensional generalized flag manifolds. Adv. Math., 185(10), 347–369.
@article{mare06,
title = {Quantum cohomology of the infinite dimensional generalized flag manifolds},
author = {Mare, A.-L.},
journal = {Adv. Math.},
volume = {185},
number = {10},
pages = {347 - 369},
year = {2004},
eprint = {math/0105133v4},
archiveprefix = {arXiv},
primaryclass = {math},
doi = {10.1016/j.aim.2003.07.005}
}
Consider the infinite dimensional flag manifold LK/T corresponding to the simple Lie group K of rank l and with maximal torus T. We show that, for K of type A, B or C, if we endow the space H^*(LK/T)⊗ ℝ[q_1,...,q_l+1] (where q_1,...,q_l+1 are multiplicative variables) with an ℝ[{q_j}]-bilinear product satisfying some simple properties analogous to the quantum product on QH^∗(K/T), then the isomorphism type of the resulting ring is determined by the integrals of motion of a certain periodic Toda lattice system, in exactly the same way as the isomorphism type of QH^∗(K/T) is determined by the integrals of motion of the non-periodic Toda lattice (see the theorem of Kim). This is a generalization of a theorem of Guest and Otofuji.
Mare, A.-L. (2004). Relations in the quantum cohomology ring of G/B. Math. Res. Lett., 11(1), 35–48.
@article{mare07,
title = {Relations in the quantum cohomology ring of G/B},
author = {Mare, A.-L.},
journal = {Math. Res. Lett.},
volume = {11},
number = {1},
pages = {35 - 48},
year = {2004},
eprint = {math/0210026v2},
archiveprefix = {arXiv},
primaryclass = {math},
%doi = {10.4310/MRL.2004.v11.n1.a5},
file = {mare03.pdf}
}
The ideal of relations in the (small) quantum cohomology ring of the generalized flag manifold G/B has been determined by B. Kim. We are going to point out a limited number of properties that, if they are satisfied by an ℝ[q1,...,q_l]-bilinear product ∘ on H^∗(G/B)⊗ℝ[q1,...,q_l], then the ring (H^∗(G/B)⊗ℝ[q1,...,ql],∘) is isomorphic to Kim’s ring.
Jeffrey, L. C., & Mare, A.-L. (2003). The kernel of the equivariant Kirwan map and the residue formula. Quart. J. Math., 54(4), 435–444.
@article{mare08,
title = {The kernel of the equivariant Kirwan map and the residue formula},
author = {Jeffrey, L. C. and Mare, A.-L.},
journal = {Quart. J. Math.},
volume = {54},
number = {4},
pages = {435 - 444},
year = {2003},
eprint = {math/0211119v1},
archiveprefix = {arXiv},
primaryclass = {math},
doi = {10.1093/qmath/hag026}
}
Using the notion of equivariant Kirwan map, as defined by Goldin, we prove that – in the case of Hamiltonian torus actions with isolated fixed points – Tolman and Weitsman’s description of the kernel of the Kirwan map can be deduced directly from the residue theorem of Jeffrey and Kirwan. A characterization of the kernel of the Kirwan map in terms of residues of one variable (i.e. associated to Hamiltonian circle actions) is obtained.
Mare, A.-L. (2002). Polynomial representatives of Schubert classes in QH^*(G/B). Math. Res. Lett., 9, 757–769.
We show how the quantum Chevalley formula for G/B, as stated by Peterson and proved rigorously by Fulton and Woodward, combined with ideas of Fomin, S.I. Gelfand, and Postnikov, leads to a formula which describes polynomial representatives of the Schubert cohomology classes in the canonical presentation of QH^∗(G/B) in terms of generators and relations. We generalize in this way results of Fomin, Gelfand, and Postnikov.
Mare, A.-L. (2001). Cohomology of isoparametric hypersurfaces in Hilbert space. Geom. Dedicata, 85, 21–43.
@article{mare10,
title = {Cohomology of isoparametric hypersurfaces in Hilbert space},
author = {Mare, A.-L.},
journal = {Geom. Dedicata},
volume = {85},
%number = {10},
pages = {21 - 43},
year = {2001},
doi = {10.1023/A:1010336521152},
%file = {mare04.pdf}
}
One obtains descriptions of the cohomology ring of the manifolds mentioned in the title in terms of their multiplicities and the Euler, respectively Stiefel-Whitney classes of the curvature distributions. Lifts of equifocal hypersurfaces in symmetric spaces are also discussed.
Mare, A.-L. (1997). On two results of Singhof. Comment. Math. Univ. Carolin., 38(2), 379–383.
@article{mare11,
title = {On two results of Singhof},
author = {Mare, A.-L.},
journal = {Comment. Math. Univ. Carolin.},
volume = {38},
number = {2},
pages = {379 - 383},
year = {1997},
%doi = {10.1023/A:1010336521152},
%file = {mare06.pdf},
link = {https://dml.cz/bitstream/handle/10338.dmlcz/118937/CommentatMathUnivCarolRetro_38-1997-2_20.pdf}
}
For a compact connected semisimple Lie group G we shall prove two results
(both related to a paper by Singhof) on the Lusternik-Schnirelmann category of the
adjoint orbits of G, respectively the 1-dimensional relative category of a maximal torus T in G. The techniques will be classical, but we shall also apply some basic results concerning the so-called A-category.
Conference proceedings
Jeffrey, L. C., & Mare, A.-L. (2011). On the image of real loci of symplectic manifolds under moment maps. Oberwolfach Reports, No. 09/2011, pp. 471-474.
@inproceedings{example_proceedinl,
author = {Jeffrey, L. C. and Mare, A.-L.},
year = {2011},
title = {On the image of real loci of symplectic manifolds under moment maps},
booktitle = {Oberwolfach Reports, No. 09/2011, pp. 471-474},
address = {Pasadena, CA},
doi = {10.4171/OWR/2011/09},
%file = {mare07.pdf}
}
Assume that M is a symplectic manifold acted on by a torus T in Hamiltonian fashion, the moment mapping being Φ:M→t^*. Let also τ be an anti-symplectic involutive automorphism of M which is compatible with the torus action. By a theorem of Hilgert, Neeb, and Plank, if Φ is a proper map, then the fixed point set M^τ and M have the same image under Φ. In this paper we start exploring situations when this property holds for a larger class of Hamiltonian torus actions on symplectic manifolds.
Mare, A.-L. (2002). On the theorem of Kim concerning QH^*(G/B). Integrable Systems, Topology and Physics, eds. M. Guest, R. Miyaoka, Y. Ohnita, Contemp. Math. 309, A.M.S. (2002), pp. 151-163.
@inproceedings{example_proceeding,
author = {Mare, A.-L.},
year = {2002},
title = {On the theorem of Kim concerning QH^*(G/B)},
booktitle = {Integrable Systems, Topology and Physics, eds. M. Guest, R. Miyaoka, Y. Ohnita, Contemp. Math. 309, A.M.S. (2002), pp. 151-163},
%address = {Pasadena, CA},
%file = {mare05.pdf},
doi = {10.1090/conm/309/05346}
}
A theorem of Kim gives an explicit relationship between the quantum cohomology of the generalized complex flag manifold G/B and the conserved quantities of the Toda lattice associated to the dual group (with respect to the root system) of G. The goal of this note is to outline the main ideas of Kim’s proof. The only novelty concerns the presentation of the result, namely that from the point of view of Toda lattice theory one uses the approach described by R. Goodman and N. Wallach in a series of papers published in the early 80s. Although equivalent to the descriptions of other authors, their way of defining and studying the quantum first integrals of the Toda lattice allows a more transparent connection with quantum cohomology.
Andrica, D., & Mare, L. (1998). An inequality concerning symmetric functions and some applications. Recent progress in inequalities.
Selected papers from the International Memorial Conference Dedicated to Professor Dragoslav S. Mitrinović (held in Niš, 1996), ed. G. V. Milovanović, Kluwer Academic Publishers, Dordrecht, 1998, pp. 425 - 431.
@inproceedings{example_proceedinj,
author = {Andrica, D. and Mare, L.},
year = {1998},
title = {An inequality concerning symmetric functions and some applications},
booktitle = {Recent progress in inequalities.
Selected papers from the International Memorial Conference Dedicated to Professor Dragoslav S. Mitrinović (held in Niš, 1996), ed. G. V. Milovanović, Kluwer Academic Publishers, Dordrecht, 1998, pp. 425 - 431},
address = {``Babeş-Bolyai'' University, Faculty of Mathematics and Computer Science, Cluj-Napoca, 1993},
doi = {10.1007/978-94-015-9086-0},
file = {Proceedings-inequalities.pdf}
}
A general class of inequalities concerning a symmetric function E:I^n→ℝ is obtained, I⊂ℝ being an open interval. Various inequalities involving the generalized arithmetic, geometric, and harmonic means are deduced from it.
Mare, L. (1994). On the critical set of a Morse function. Proceedings of the 23rd Conference on Geometry and Topology (held in Cluj-Napoca, 1993), eds. P. Enghis and D. Andrica, “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Cluj-Napoca, 1994, pp. 88-92.
@inproceedings{example_proceedinh,
author = {Mare, L.},
year = {1994},
title = {On the critical set of a Morse function},
booktitle = {Proceedings of the 23rd Conference on Geometry and Topology (held in Cluj-Napoca, 1993), eds. P. Enghis and D. Andrica, ``Babeş-Bolyai'' University, Faculty of Mathematics and Computer Science, Cluj-Napoca, 1994, pp. 88-92},
address = {``Babeş-Bolyai'' University, Faculty of Mathematics and Computer Science, Cluj-Napoca, 1993},
file = {Proceedings-geometry-final3.pdf}
}
We show that a necessary and sufficient condition for a (finite) subset A of a closed manifold M to be the critical set of a Morse function on M is that the cardinality of A is at least equal to the Morse number (aka the Morse-Smale characteristic) of M and is congruent mod 2 with the Euler-Poincaré characteristic of M.
Mare, L. (1993). On a property of the exponential image. Proceedings of Symposium in Geometry (held in Cluj-Napoca and Târgu Mures, 1992), eds. P. Enghis and D. Andrica, “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Cluj-Napoca, 1993, pp. 127 - 132.
@inproceedings{example_proceedini,
author = {Mare, L.},
year = {1993},
title = {On a property of the exponential image},
booktitle = {Proceedings of Symposium in Geometry (held in Cluj-Napoca and T\^argu Mures, 1992), eds. P. Enghis and D. Andrica, ``Babeş-Bolyai'' University, Faculty of Mathematics and Computer Science, Cluj-Napoca, 1993, pp. 127 - 132},
address = {``Babeş-Bolyai'' University, Faculty of Mathematics and Computer Science, Cluj-Napoca, 1993},
file = {Proceedings-exponential-merged.pdf}
}
It is known that the image of the exponential map on a semisimple complex Lie group G is a topologically dense subset of the group. If G is real, the image of the exponential map is in general not topologically dense (also not closed or open). In this note we show if G is a real semismple Lie group then the image of the exponential map of G satisfies the following property: its interior is a dense subspace of it. The proof relies on a result proved by the author in a separate note, entitled "On a linear matrix operator", which is appended to the paper.
Mare, L. (1989). Asupra une note din Gazeta Matematică (in Romanian). Lucrările Seminarului “Didactica Matematicii", vol. 5, Universitatea din Cluj-Napoca, Facultatea de Matematică și Fizică, 1989, pp. 165–172.
@inproceedings{example_proceedink,
author = {Mare, L.},
year = {1989},
title = {Asupra une note din Gazeta Matematică (in Romanian)},
booktitle = {Lucrările Seminarului ``Didactica Matematicii", vol. 5, Universitatea din Cluj-Napoca, Facultatea de Matematică și Fizică, 1989, pp. 165–172},
file = {Proceedings-Didactica.pdf}
}
To any polynomial f with coefficients in ℂ one can canonically associate a map from the space M_n(ℂ) of nxn square matrices to itself, n≥2. In this note the question of surjectivity of this map is addressed: the question was initially raised by D. Andrica in a paper published in Gazeta Matematică. In this paper we prove that the aforementioned surjectivity property holds iff for any α∈ℂ the equation f(z)=α has at least one simple root. An example is the polynomial 1+1/1!z+1/2!z^2+...+1/p!z^p, where p is an odd integer. The case when ℂ is replaced by ℝ is also touched upon, being more difficult to deal with.
Others
Mare, A.-L. (1998). Topology of isotropy orbits, Ph.D. Thesis, University of Augsburg, 1998. in Wissner-Verlag, Augsburger mathematisch-naturwissenschaftliche Schriften, 25.
@inothers{example_proceedinm,
author = {Mare, A.-L.},
year = {1998},
title = {Topology of isotropy orbits, Ph.D. Thesis, University of Augsburg, 1998},
booktitle = { Wissner-Verlag, Augsburger mathematisch-naturwissenschaftliche Schriften, 25},
link = {https://d-nb.info/953888827},
address = {Pasadena, CA},
%file = {Teza.pdf}
}
Isotropy orbits of semisimple non-compact symmetric spaces are investigated from the point of view of their homology groups and even cohomology algebras. Also known under the name of "real flag manifolds", their topology was initially investigated by R. Bott and H. Samelson in the late 50s and then further by W.-Y. Hsiang, R. Palais, and C.-L. Terng in a work published in 1988. As pointed out in the latter article, it is convenient to view the orbits as special cases of elements of isoparametric foliations of euclidean space. This viewpoint was also adopted in the Thesis. Informations concerning the topology of isoparametric submanifolds and their focal manifolds can be naturally obtained by using their embedding in euclidean space and the corresponding height functions. These functions are all of Morse-Bott type and can thus readily provide information about the homology groups. Some refinements of the classical Morse theory are needed in order to get information about the cohomology algebra. Namely, one can construct so-called Bott-Samelson cycles. One of the main achievements of the Thesis is a construction of a new version of such cycles, which simplifies in some repects the computations of the cohomology. In the special case of Grassmann manifolds, one even obtains a relationship between the aforementioned cycles and the Schubert cycles.