@article{helminck_helminck_2024, title={A construction of solutions of an integrable deformation of a commutative Lie algebra of skew hermitian Z×Z-matrices}, url={https://doi.org/10.1016/j.indag.2024.04.001}, DOI={10.1016/j.indag.2024.04.001}, abstractNote={Inside the algebra LTZ(R) of Z×Z-matrices with coefficients from a commutative ℂ-algebra R that have only a finite number of nonzero diagonals above the central diagonal, we consider a deformation of a commutative Lie algebra Csh(ℂ) of finite band skew hermitian matrices that is different from the Lie subalgebras that were deformed at the discrete KP hierarchy and its strict version. The evolution equations that the deformed generators of Csh(ℂ) have to satisfy are determined by the decomposition of LTZ(R) in the direct sum of an algebra of lower triangular matrices and the finite band skew hermitian matrices. This yields then the Csh(ℂ)-hierarchy. We show that the projections of a solution satisfy zero curvature relations and that it suffices to solve an associated Cauchy problem. Solutions of this type can be obtained by finding appropriate vectors in the LTZ(R)-module of oscillating matrices, the so-called wave matrices, that satisfy a set of equations in the oscillating matrices, called the linearization of the Csh(ℂ)-hierarchy. Finally, a Hilbert Lie group will be introduced from which wave matrices for the Csh(ℂ)-hierarchy are constructed. There is a real analogue of the Csh(ℂ)-hierarchy called the Cas(R)-hierarchy. It consists of a deformation of a commutative Lie algebra Cas(R) of anti-symmetric matrices. We will properly introduce it here too on the way and mention everywhere the corresponding result for this hierarchy, but we leave its proofs mostly to the reader.}, journal={Indagationes Mathematicae}, author={Helminck, Aloysius G. and Helminck, Gerardus F.}, year={2024}, month={Apr} }
@article{buell_helminck_klima_schaefer_wright_ziliak_2020, title={Classifying the orbits of the generalized symmetric spaces for}, volume={48}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85078488640&partnerID=MN8TOARS}, DOI={10.1080/00927872.2019.1705471}, abstractNote={In this paper we will discuss the orbits of the fixed-point group on the tori of the generalized symmetric spaces of SL2(k) where k is a finite field. Specifically, we will provide a characterization and classification of the maximal k-split and k-anisotropic tori.}, number={4}, journal={Communications in Algebra}, author={Buell, C. and Helminck, A. and Klima, V. and Schaefer, J. and Wright, C. and Ziliak, E.}, year={2020}, pages={1744–1757} }
@article{collins_haas_helminck_lenarz_pelatt_saccon_welz_2020, title={Extended symmetric spaces and θ-twisted involution graphs}, volume={48}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85078902156&partnerID=MN8TOARS}, DOI={10.1080/00927872.2019.1711106}, abstractNote={For a Weyl group G and an automorphism θ of order 2, the set of involutions and θ-twisted involutions can be generated by considering actions by basis elements, creating a poset structure on the elements. Haas and Helminck showed that there is a relationship between these sets and their Bruhat posets. We extend that result by considering other bases and automorphisms. We show for G = Sn, θ an involution, and any basis consisting of transpositions, the extended symmetric space is generated by a similar algorithm. Moreover, there is an isomorphism of the poset graphs for certain bases and θ.}, number={6}, journal={Communications in Algebra}, author={Collins, J.B. and Haas, R. and Helminck, A.G. and Lenarz, J. and Pelatt, K.E. and Saccon, S. and Welz, M.}, year={2020}, pages={2293–2306} }
@article{buell_helminck_klima_schaefer_wright_ziliak_2017, title={On the structure of generalized symmetric spaces of SL2(Fq) and GL2(Fq)}, volume={37}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85042125628&partnerID=MN8TOARS}, DOI={10.1285/i15900932v37n2p1}, abstractNote={In this paper we will discuss the generalized symmetric spaces for and . Specifically we will characterize the structure of these spaces and prove that when the characteristic of is not equal to two the extended generalized symmetric space is equal to the generalized symmetric space for and nearly equal for for all but one involution.}, number={2}, journal={Note di Matematica}, author={Buell, C. and Helminck, A.G. and Klima, V. and Schaefer, J. and Wright, C. and Ziliak, E.}, year={2017}, pages={1–10} }
@article{benim_helminck_ward_2016, title={Erratum of "Isomorphy classes of involutions of SP(2n, k), n > 2"}, volume={26}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84955487887&partnerID=MN8TOARS}, number={1}, journal={Journal of Lie Theory}, author={Benim, R.W. and Helminck, A.G. and Ward, F.J.}, year={2016}, pages={293–295} }
@article{benim_dometrius_helminck_wu_2016, title={Isomorphy classes of involutions of SO(n, k, beta), n > 2}, volume={26}, number={2}, journal={Journal of Lie Theory}, author={Benim, R. W. and Dometrius, C. E. and Helminck, A. G. and Wu, L.}, year={2016}, pages={383–438} }
@article{benim_dometrius_helminck_wu_2016, title={Isomorphy classes of involutions of SO(n, κ, β), n > 2}, volume={26}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84983329156&partnerID=MN8TOARS}, number={2}, journal={Journal of Lie Theory}, author={Benim, R.W. and Dometrius, C.E. and Helminck, A.G. and Wu, L.}, year={2016}, pages={383–438} }
@article{benim_helminck_ward_2016, title={Isomorphy classes of involutions of SP(2n, k), n > 2 (vol 25, pg 903, 2015)}, volume={26}, number={1}, journal={Journal of Lie Theory}, author={Benim, R. W. and Helminck, A. G. and Ward, F. J.}, year={2016}, pages={293–295} }
@article{benim_helminck_ward_2015, title={Isomorphy classes of involutions of SP(2n, k), n > 2}, volume={25}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84929886770&partnerID=MN8TOARS}, number={4}, journal={Journal of Lie Theory}, author={Benim, R.W. and Helminck, A.G. and Ward, F.J.}, year={2015}, pages={903–947} }
@article{helminck_helminck_panasenko_2014, title={Cauchy problems related to integrable deformations of pseudo differential operators}, volume={85}, ISSN={["1879-1662"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84953206394&partnerID=MN8TOARS}, DOI={10.1016/j.geomphys.2014.05.004}, abstractNote={In this paper we discuss the solvability of two Cauchy problems in the pseudo differential operators. The first is associated with a set of pseudo differential operators of negative order, the prominent example being the set of strict integral operator parts of the different powers of a solution of the KP hierarchy. We show that it can be solved, provided the setting possesses a compatibility completeness. In such a setting all solutions of the KP hierarchy are obtained by dressing with the solution of the related Cauchy problem. The second Cauchy problem is slightly more general and links up with a set of pseudo differential operators of order zero or less. The key example here is the collection of integral operator parts of the different powers of a solution of the strict KP hierarchy. This system is solvable as soon as exponential and compatibility completeness holds. Also under these circumstances, all solutions of the strict KP hierarchy are obtained by dressing with the solution of the corresponding Cauchy problem.}, journal={JOURNAL OF GEOMETRY AND PHYSICS}, author={Helminck, G. F. and Helminck, A. G. and Panasenko, E. A.}, year={2014}, month={Nov}, pages={196–205} }
@article{helminck_helminck_2014, title={Infinite dimensional symmetric spaces and Lax equations compatible with the infinite Toda chain}, volume={85}, ISSN={["1879-1662"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84955182312&partnerID=MN8TOARS}, DOI={10.1016/j.geomphys.2014.05.023}, abstractNote={In this paper we present a natural embedding of the infinite Toda chain in a set of Lax equations in the algebra L T consisting of Z × Z -matrices that possess only a finite number of nonzero diagonals above the main central diagonal. This hierarchy of Lax equations describes the evolution of deformations of a set of commuting anti-symmetric matrices and corresponds to splitting this algebra into its anti-symmetric part and the subalgebra of matrices in L T that have no component above the main diagonal. We show that the projections of these deformations satisfy a set of zero curvature relations, which demonstrates the compatibility of the system. Further we introduce a suitable L T -module in which we can distinguish elements, the so-called wave matrices, that will lead you to solutions of the hierarchy. We conclude by showing how wave matrices of the infinite Toda chain hierarchy can be constructed starting from an infinite dimensional symmetric space.}, journal={JOURNAL OF GEOMETRY AND PHYSICS}, author={Helminck, G. F. and Helminck, A. G.}, year={2014}, month={Nov}, pages={60–74} }
@article{cunningham_edgar_helminck_jones_oh_schwell_vasquez_2014, title={On the structure of involutions and symmetric spaces of dihedral groups}, volume={34}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84923120162&partnerID=MN8TOARS}, DOI={10.1285/i15900932v34n2p23}, number={2}, journal={Note di Matematica}, author={Cunningham, K.K.A. and Edgar, T. and Helminck, A.G. and Jones, B.F. and Oh, H. and Schwell, R. and Vasquez, J.F.}, year={2014}, pages={23–40} }
@article{helminck_helminck_panasenko_2013, title={Integrable deformations in the algebra of pseudodifferential operators from a Lie algebraic perspective}, volume={174}, ISSN={["0040-5779"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84873586362&partnerID=MN8TOARS}, DOI={10.1007/s11232-013-0011-7}, number={1}, journal={THEORETICAL AND MATHEMATICAL PHYSICS}, author={Helminck, G. F. and Helminck, A. G. and Panasenko, E. A.}, year={2013}, month={Jan}, pages={134–153} }
@article{haas_helminck_2012, title={Algorithms for Twisted Involutions in Weyl Groups}, volume={19}, ISSN={["1005-3867"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84859536990&partnerID=MN8TOARS}, DOI={10.1142/s100538671200017x}, abstractNote={ Let (W, Σ) be a finite Coxeter system, and θ an involution such that θ (Δ) = Δ, where Δ is a basis for the root system Φ associated with W, and [Formula: see text] the set of θ-twisted involutions in W. The elements of [Formula: see text] can be characterized by sequences in Σ which induce an ordering called the Richardson-Spinger Bruhat poset. The main algorithm of this paper computes this poset. Algorithms for finding conjugacy classes, the closure of an element and special cases are also given. A basic analysis of the complexity of the main algorithm and its variations is discussed, as well experience with implementation. }, number={2}, journal={ALGEBRA COLLOQUIUM}, author={Haas, R. and Helminck, A. G.}, year={2012}, month={Jun}, pages={263–282} }
@article{cahn_haas_helminck_li_schwartz_2012, title={Permutation notations for the exceptional Weyl groupF4}, volume={5}, url={http://dx.doi.org/10.2140/involve.2012.5.81}, DOI={10.2140/involve.2012.5.81}, abstractNote={This paper describes a permutation notation for the Weyl groups of type F 4 and G 2 .The image in the permutation group is presented as well as an analysis of the structure of the group.This description enables faster computations in these Weyl groups which will prove useful for a variety of applications.}, number={1}, journal={Involve, a Journal of Mathematics}, publisher={Mathematical Sciences Publishers}, author={Cahn, Patricia and Haas, Ruth and Helminck, Aloysius and Li, Juan and Schwartz, Jeremy}, year={2012}, month={Apr}, pages={81–89} }
@article{haas_helminck_2011, title={Admissible Sequences for Twisted Involutions in Weyl Groups}, volume={54}, ISSN={["0008-4395"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84859551012&partnerID=MN8TOARS}, DOI={10.4153/cmb-2011-075-1}, abstractNote={AbstractLetW be a Weyl group, Σ a set of simple reflections inW related to a basis Δ for the root system Φ associated with W and θ an involution such that θ(Δ) = Δ. We show that the set of θ- twisted involutions in W, = {w ∈ W | θ(w) = w–1} is in one to one correspondence with the set of regular involutions . The elements of are characterized by sequences in Σ which induce an ordering called the Richardson–Springer Poset. In particular, for Φ irreducible, the ascending Richardson–Springer Poset of , for nontrivial θ is identical to the descending Richardson–Springer Poset of .}, number={4}, journal={CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES}, author={Haas, Ruth and Helminck, Aloysius G.}, year={2011}, month={Dec}, pages={663–675} }
@article{helminck_krasil’shchik_rubtsov_2011, title={Dedication to Gerardus F. Helminck}, volume={61}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-79959805139&partnerID=MN8TOARS}, DOI={10.1016/j.geomphys.2011.04.003}, abstractNote={HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.L'archive ouverte pluridisciplinaire}, number={9}, journal={Journal of Geometry and Physics}, author={Helminck, A.G. and Krasil’shchik, J. and Rubtsov, V.}, year={2011}, pages={1631} }
@article{helminck_krasil’shchik_rubtsov_2011, title={Editors' preface for the topical issue on "The interface between integrability and quantization"}, volume={61}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-79959786220&partnerID=MN8TOARS}, DOI={10.1016/j.geomphys.2011.04.002}, abstractNote={published or not.The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.}, number={9}, journal={Journal of Geometry and Physics}, author={Helminck, A.G. and Krasil’shchik, J. and Rubtsov, V.}, year={2011}, pages={1632} }
@article{helminck_helminck_opimakh_2011, title={Equivalent forms of multi component Toda hierarchies (Reprinted from JOURNAL OF GEOMETRY AND PHYSICS, vol 61, pg 847, 2011)}, volume={61}, ISSN={["1879-1662"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-79959800191&partnerID=MN8TOARS}, DOI={10.1016/j.geomphys.2011.06.012}, abstractNote={In this paper we consider various sets of commuting directions in the Z×Z-matrices. For each k≥1, we decompose the Z×Z-matrices in k×k-blocks. The set of basic commuting directions splits then roughly speaking half in a set of directions that are upper triangular w.r.t. this decomposition and half in a collection of directions that possess a lower triangular form. Next we consider deformations of each set in respectively the upper k×k-block triangular Z×Z-matrices and the strictly lower k×k-block triangular Z×Z-matrices that preserve the commutativity of the generators of each subset and for which the evolution w.r.t. the parameters of the opposite set is compatible. It gives rise to an integrable hierarchy consisting of a set of evolution equations for the perturbations of the basic directions. They amount to a tower of differential and difference equations for the coefficients of these perturbed matrices. The equations of the hierarchy are conveniently formulated in so-called Lax equations for these perturbations. They possess a minimal realization for which it is shown that the relevant evolutions of the perturbation commute. These Lax equations are shown in a purely algebraic way to be equivalent to zero curvature equations for a collection of finite band matrices. As the name zero curvature equations suggests there is a Cauchy problem related to these equations. Therefore a description of the relevant infinite Cauchy problems is given together with a discussion of its solvability and uniqueness. There exists still another form of the nonlinear equations of the hierarchy: the bilinear form. It requires the notion of wave matrices and a description of the related linearizations and then we can show how this bilinear form is equivalent with the Lax form. We conclude with the construction of solutions of the hierarchy.}, number={9}, journal={JOURNAL OF GEOMETRY AND PHYSICS}, author={Helminck, G. F. and Helminck, A. G. and Opimakh, A. V.}, year={2011}, month={Sep}, pages={1755–1781} }
@article{helminck_schwarz_2011, title={On generalized Cartan subspaces}, volume={16}, ISSN={["1083-4362"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-80051892373&partnerID=MN8TOARS}, DOI={10.1007/s00031-011-9151-8}, number={3}, journal={TRANSFORMATION GROUPS}, author={Helminck, Aloysius G. and Schwarz, Gerald W.}, year={2011}, month={Sep}, pages={783–805} }
@article{helminck_helminck_opimakh_2011, title={Equivalent forms of multi component Toda hierarchies}, volume={61}, ISSN={["1879-1662"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-78651407120&partnerID=MN8TOARS}, DOI={10.1016/j.geomphys.2010.11.013}, abstractNote={In this paper we consider various sets of commuting directions in the Z×Z-matrices. For each k≥1, we decompose the Z×Z-matrices in k×k-blocks. The set of basic commuting directions splits then roughly speaking half in a set of directions that are upper triangular w.r.t. this decomposition and half in a collection of directions that possess a lower triangular form. Next we consider deformations of each set in respectively the upper k×k-block triangular Z×Z-matrices and the strictly lower k×k-block triangular Z×Z-matrices that preserve the commutativity of the generators of each subset and for which the evolution w.r.t. the parameters of the opposite set is compatible. It gives rise to an integrable hierarchy consisting of a set of evolution equations for the perturbations of the basic directions. They amount to a tower of differential and difference equations for the coefficients of these perturbed matrices. The equations of the hierarchy are conveniently formulated in so-called Lax equations for these perturbations. They possess a minimal realization for which it is shown that the relevant evolutions of the perturbation commute. These Lax equations are shown in a purely algebraic way to be equivalent to zero curvature equations for a collection of finite band matrices. As the name zero curvature equations suggests there is a Cauchy problem related to these equations. Therefore a description of the relevant infinite Cauchy problems is given together with a discussion of its solvability and uniqueness. There exists still another form of the nonlinear equations of the hierarchy: the bilinear form. It requires the notion of wave matrices and a description of the related linearizations and then we can show how this bilinear form is equivalent with the Lax form. We conclude with the construction of solutions of the hierarchy.}, number={4}, journal={JOURNAL OF GEOMETRY AND PHYSICS}, author={Helminck, G. F. and Helminck, A. G. and Opimakh, A. V.}, year={2011}, month={Apr}, pages={847–873} }
@article{helminck_helminck_opimakh_2010, title={THE RELATIVE FRAME BUNDLE OF AN INFINITE-DIMENSIONAL FLAG VARIETY AND SOLUTIONS OF INTEGRABLE HIERARCHIES}, volume={165}, ISSN={["0040-5779"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-79151472227&partnerID=MN8TOARS}, DOI={10.1007/s11232-010-0133-0}, number={3}, journal={THEORETICAL AND MATHEMATICAL PHYSICS}, author={Helminck, G. F. and Helminck, A. G. and Opimakh, A. V.}, year={2010}, month={Dec}, pages={1610–1636} }
@article{helminck_2010, title={On Orbit Decompositions for Symmetric k-Varieties}, volume={278}, ISBN={["978-0-8176-4874-9"]}, ISSN={["0743-1643"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85028334984&partnerID=MN8TOARS}, DOI={10.1007/978-0-8176-4875-6_6}, abstractNote={Orbit decompositions play a fundamental role in the study of symmetric k-varieties and their applications to representation theory and many other areas of mathematics, such as geometry, the study of automorphic forms and character sheaves. Symmetric k-varieties generalize symmetric varieties and are defined as the homogeneous spaces G k /H k , where G is a connected reductive algebraic group defined over a field k of characteristic not 2, H the fixed point group of an involution σ and G k (resp., H k ) the set of k-rational points of G (resp., H). In this contribution we give a survey of results on the various orbit decompositions which are of importance in the study of these symmetric k-varieties and their applications with an emphasis on orbits of parabolic k-subgroups acting on symmetric k-varieties. We will also discuss a number of open problems.}, journal={SYMMETRY AND SPACES}, publisher={Birkhäuser Boston, Boston, MA}, author={Helminck, A. G.}, year={2010}, pages={83–127} }
@article{beun_helminck_2009, title={On the Classification of Orbits of Symmetric Subgroups Acting on Flag Varieties of SL(2, k)}, volume={37}, ISSN={["1532-4125"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-69249248124&partnerID=MN8TOARS}, DOI={10.1080/00927870802466983}, abstractNote={Symmetric k-varieties are a generalization of symmetric spaces to general fields. Orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential in the study of symmetric k-varieties and their representations. In this article, we present the classification of these orbits for the group SL(2,k) for a number of base fields k, including finite fields and the 𝔭-adic numbers. We use the characterization in Helminck and Wang (1993), which requires one to first classify the orbits of the θ-stable maximal k-split tori under the action of the k-points of the fixed point group.}, number={4}, journal={COMMUNICATIONS IN ALGEBRA}, author={Beun, Stacy L. and Helminck, Aloysius G.}, year={2009}, pages={1334–1352} }
@article{helminck_schwarz_2009, title={Real double coset spaces and their invariants}, volume={322}, ISSN={["1090-266X"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-67349157728&partnerID=MN8TOARS}, DOI={10.1016/j.jalgebra.2009.01.028}, abstractNote={Let G be a real form of a complex reductive group. Suppose that we are given involutions σ and θ of G. Let H=Gσ denote the fixed group of σ and let K=Gθ denote the fixed group of θ. We are interested in calculating the double coset space H\G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles.}, number={1}, journal={JOURNAL OF ALGEBRA}, author={Helminck, Aloysius G. and Schwarz, Gerald W.}, year={2009}, month={Jul}, pages={219–236} }
@article{daniel_helminck_2008, title={Algorithms for computations in local symmetric spaces}, volume={36}, ISSN={["1532-4125"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-45949095294&partnerID=MN8TOARS}, DOI={10.1080/00927870801940434}, abstractNote={In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma, and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this article we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties.}, number={5}, journal={COMMUNICATIONS IN ALGEBRA}, author={Daniel, Jennifer R. and Helminck, Aloysius G.}, year={2008}, month={May}, pages={1758–1788} }
@article{gagliardi_helminck_2007, title={Algorithms for computing characters for symmetric spaces}, volume={99}, ISSN={["1572-9036"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-36049006830&partnerID=MN8TOARS}, DOI={10.1007/s10440-007-9171-5}, number={3}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Gagliardi, Daniel and Helminck, Aloysius G.}, year={2007}, month={Dec}, pages={339–365} }
@article{daniel_helminck_2007, title={Computing the fine structure of real reductive symmetric spaces}, volume={42}, ISSN={["0747-7171"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-34247628477&partnerID=MN8TOARS}, DOI={10.1016/j.jsc.2006.08.003}, abstractNote={Much of the structure of Lie groups has been implemented in several computer algebra packages, including LiE , GAP4, Chevie, Magma and Maple. The structure of reductive symmetric spaces is very similar to that of the underlying Lie group and a computer algebra package for computations related to symmetric spaces would be an important tool for researchers in many areas of mathematics. Until recently only very few algorithms existed for computations in symmetric spaces due to the fact that their structure is much more complicated than that of the underlying group. In recent work, Daniel and Helminck [Daniel, J.R., Helminck, A.G., 2004. Algorithms for computations in local symmetric spaces. Comm. Algebra (in press)] gave a complete set of algorithms for computing the fine structure of Riemannian symmetric spaces. In this paper we make the first step in extending these results to general real reductive symmetric spaces and give a number of algorithms for computing some of their fine structure. This case is a lot more complicated since it involves the intricate relations of five root systems and their Weyl groups instead of just two as in the Riemannian case. We show first that this fine structure can be obtained from the setting of a complex reductive Lie group with a pair of commuting involutions. Then we proceed to give a number of algorithms for computing the fine structure of the latter.}, number={5}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Daniel, Jennifer R. and Helminck, Aloysius G.}, year={2007}, month={May}, pages={497–510} }
@article{haas_helminck_rizki_2007, title={Properties of twisted involutions in signed permutation notation}, volume={62}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-78651544802&partnerID=MN8TOARS}, journal={Journal of Combinatorial Mathematics and Combinatorial Computing}, author={Haas, R. and Helminck, A.G. and Rizki, N.}, year={2007}, pages={121–128} }
@inproceedings{gagliardi_helminck_2006, title={Implementation of algorithms for computing characters for
symmetric spaces}, volume={180}, booktitle={Congr. Numer.}, author={Gagliardi, Daniel and Helminck, Aloysius G.}, year={2006}, pages={5–20} }
@inproceedings{brenneman_haas_helminck_2006, title={Implementing an algorithm for the twisted involution poset for
Weyl groups}, volume={182}, booktitle={Congr. Numer.}, author={Brenneman, Kathryn and Haas, Ruth and Helminck, Aloysius G.}, year={2006}, pages={137–144} }
@article{helminck_wu_dometrius_2006, title={Involutions of SL(n,k), (n < 2)}, volume={90}, ISSN={["1572-9036"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-33745726961&partnerID=MN8TOARS}, DOI={10.1007/s10440-006-9032-7}, number={1-2}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Helminck, Aloysius G. and Wu, Ling and Dometrius, Christopher E.}, year={2006}, month={Jan}, pages={91–119} }
@article{helminck_helminck_2005, title={Multiplicity one for representations corresponding to spherical distribution vectors of class rho}, volume={86}, ISSN={["1572-9036"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-17444405320&partnerID=MN8TOARS}, DOI={10.1007/s10440-005-0461-5}, abstractNote={In this paper one considers a unimodular second countable locally compact group G and the homogeneous space X:=H\G, where H is a closed unimodular subgroup of G. Over X complex vector bundles are considered such that H acts on the fibers by a unitary representation ρ with closed image. The natural action of G on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class ρ. The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products. In the case that H is compact, the multiplicity free decomposition is shown to be equivalent with the commutativity of a suitable convolution algebra. As an example, one takes for X a symmetric k-variety $\mathcal{H}_{k}\backslash \mathcal{G}_{k}$ , with k a locally compact field of characteristic not equal to two, and for ρ a character of ℋk, whose square is trivial. Here $\mathcal{G}$ is a reductive algebraic group defined over k and ℋ is the fixed point group of an involution σ of $\mathcal{G}$ defined over k. It is shown then that the natural representation ℒ of G k on the Hilbert space $L^{2}(\mathcal{H}_{k}\backslash \mathcal{G}_{k})$ is multiplicity free if ℋ is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the non-Archimedean case, a general procedure is given that might lead to showing that a pair $(\mathcal{G}_{k},\mathcal{H}_{k})$ is a generalized Gelfand pair. Here $\mathcal{G}$ and ℋ are suitable algebraic groups defined over k.}, number={1-2}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Helminck, AG and Helminck, GF}, year={2005}, month={Mar}, pages={21–48} }
@article{helminck_schwarz_2004, title={Smoothness of quotients associated with a pair of commuting involutions}, volume={56}, ISSN={["1496-4279"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-5644302657&partnerID=MN8TOARS}, DOI={10.4153/CJM-2004-043-7}, abstractNote={AbstractLet σ, θ be commuting involutions of the connected semisimple algebraic group G where σ, θ and G are defined over an algebraically closed field , char = 0. Let H := Gσ and K := Gθ be the fixed point groups. We have an action (H × K) × G → G, where ((h, k), g) ⟼ hgk–1, h ∈ H, k ∈ K, g ∈ G. Let G//(H × K) denote the categorical quotient Spec (G)H×K. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg [Ste75], Pittie [Pit72] and Richardson [Ric82] in the symmetric case where σ = θ and H = K.}, number={5}, journal={CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES}, publisher={Amer. Math. Soc., Providence, RI}, author={Helminck, AG and Schwarz, GW}, year={2004}, month={Oct}, pages={945–962} }
@article{helminck_wu_2002, title={Classification of involutions of SL(2, k)}, volume={30}, ISSN={["0092-7872"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0036003453&partnerID=MN8TOARS}, DOI={10.1081/AGB-120006486}, abstractNote={ABSTRACT In this paper we give a simple characterization of the isomorphy classes of involutions of with k any field of characteristic not 2. We also classify the isomorphy classes of involutions for k algebraically closed, the real numbers, the -adic numbers and finite fields. We determine in which cases the corresponding fixed point group H is k -anisotropic. In those cases the corresponding symmetric k -variety consists of semisimple elements.}, number={1}, journal={COMMUNICATIONS IN ALGEBRA}, author={Helminck, AG and Wu, L}, year={2002}, pages={193–203} }
@article{helminck_helminck_2002, title={Hilbert flag varieties and their Kahler structure}, volume={35}, ISSN={["0305-4470"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0037064104&partnerID=MN8TOARS}, DOI={10.1088/0305-4470/35/40/312}, abstractNote={In this paper, we introduce the infinite-dimensional flag varieties associated with integrable systems of the KdV- and Toda-type and discuss the structure of these manifolds. As an example we treat the Fubini–Study metric on the projective space associated with a separable complex Hilbert space and conclude by showing that all flag varieties introduced before possess a Kahler structure.}, number={40}, journal={JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL}, author={Helminck, GF and Helminck, AG}, year={2002}, month={Oct}, pages={8531–8550} }
@article{helminck_schwarz_2002, title={Orbits and invariants associated with a pair of spherical varieties: Some examples}, volume={73}, ISSN={["0167-8019"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0036693447&partnerID=MN8TOARS}, DOI={10.1023/A:1019726804264}, number={1-2}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Helminck, AG and Schwarz, GW}, year={2002}, month={Aug}, pages={103–113} }
@article{helminck_helminck_2002, title={Spherical distribution vectors}, volume={73}, ISSN={["1572-9036"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0036692685&partnerID=MN8TOARS}, DOI={10.1023/A:1019762302447}, abstractNote={In this paper we consider a locally compact second countable unimodular group G and a closed unimodular subgroup H. Let ρ be a finite-dimensional unitary representation of H with closed image. For the unitary representation of G obtained by inducing ρ from H to G a decomposition in Hilbert subspaces of a certain space of distributions is given. It is shown that the representations relevant for this decomposition are determined by so-called (ρ,H) spherical distributions, which leads to a description of the decomposition on the level of these distributions.}, number={1-2}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Helminck, AG and Helminck, GF}, year={2002}, month={Aug}, pages={39–57} }
@article{helminck_schwarz_2001, title={Orbits and invariants associated with a pair of commuting involutions}, volume={106}, ISSN={["0012-7094"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0013166172&partnerID=MN8TOARS}, DOI={10.1215/s0012-7094-01-10622-4}, abstractNote={Let σ, θ be commuting involutions of the connected reductive algebraic group G where σ, θ and G are defined over a (usually algebraically closed) field k, char k = 2. We have fixed point groups H := G and K := G and an action (H × K ) × G → G, where ((h, k), g) → hgk−1, h ∈ H, k ∈ K, g ∈ G. Let G//(H × K ) denote Spec O(G)H×K (the categorical quotient). Let A be maximal among subtori S of G such that θ(s) = σ(s) = s−1 for all s ∈ S. There is the associated Weyl group W := WH×K (A). We show: • The inclusion A → G induces an isomorphism A/W ∼ → G//(H × K ). In particular, the closed (H × K )-orbits are precisely those which intersect A. • The fibers of G → G//(H × K ) are the same as those occurring in certain associated symmetric varieties. In particular, the fibers consist of finitely many orbits. We investigate: • The structure of W and its relation to other naturally occurring Weyl groups and to the action of σθ on the A-weight spaces of . • The relation of the orbit type stratifications of A/W and G//(H × K ). Along the way we simplify some of Richardson’s proofs for the symmetric case σ = θ, and at the end we quickly recover results of Berger, Flensted-Jensen, Hoogenboom and Matsuki [Ber57, FJ78, Hoo84, Mat97] for the case k = .}, number={2}, journal={DUKE MATHEMATICAL JOURNAL}, author={Helminck, AG and Schwarz, GW}, year={2001}, month={Feb}, pages={237–279} }
@article{helminck_2000, title={Computing orbits of minimal parabolic k-subgroups acting on symmetric k-varieties}, volume={30}, ISSN={["0747-7171"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0013168582&partnerID=MN8TOARS}, DOI={10.1006/jsco.2000.0395}, abstractNote={In this paper we present an algorithm to compute the orbits of a minimal parabolic k -subgroup acting on a symmetric k -variety and most of the combinatorial structure of the orbit decomposition. This algorithm can be implemented in LiE, GAP4, Magma, Maple or in a separate program. These orbits are essential in the study of symmetric k -varieties and their representations. In a similar way to the special case of a Borel subgroup acting on the symmetric variety, (see A. G. Helminck. Computing B -orbits on G/H. J. Symb. Comput.,21 , 169?209, 1996.) one can use the associated twisted involutions in the restricted Weyl group to describe these orbits (see A. G. Helminck and S. P. Wang. On rationality properties of involutions of reductive groups. Adv. Math., 99, 26?96, 1993). However, the orbit structure in this case is much more complicated than the special case of orbits of a Borel subgroup. We will first modify the characterization of the orbits of minimal parabolic k -subgroups acting on the symmetric k -varieties given in Helminck and Wang (1993), to illuminate the similarity to the one for orbits of a Borel subgroup acting on a symmetric variety in Helminck (1996). Using this characterization we show how the algorithm in Helminck (1996) can be adjusted and extended to compute these twisted involutions as well.}, number={5}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Helminck, AG}, year={2000}, month={Nov}, pages={521–553} }
@article{brion_helminck_2000, title={On orbit closures of symmetric subgroups in flag varieties}, volume={52}, ISSN={["1496-4279"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0034385698&partnerID=MN8TOARS}, DOI={10.4153/CJM-2000-012-9}, abstractNote={AbstractWe study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism θ. Generalizing work of Springer, we parametrize the (finite) orbit set K \ G/P and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of θ-stable (resp. θ-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B ⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle on G/B having nonzero global sections, we show that the restriction map resX : H0(G/B, ) → H0(X, ) is surjective and that Hi(X, ) = 0 for i ≥ 1. Moreover, we describe the K-module H0(X, ). This gives information on the restriction to K of the simple G-module H0(G/B, ). Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal K-types.}, number={2}, journal={CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES}, author={Brion, M and Helminck, AG}, year={2000}, month={Apr}, pages={265–292} }
@article{helminck_2000, title={On the classification of k-involutions}, volume={153}, ISSN={["0001-8708"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0034660820&partnerID=MN8TOARS}, DOI={10.1006/aima.1998.1884}, abstractNote={Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ, and Gk (resp. Hk) the set of k-rational points of G (resp. H). The variety Gk/Hk is called a symmetric k-variety. These varieties occur in many problems in representation theory, geometry, and singularity theory. Over the last few decades the representation theory of these varieties has been extensively studied for k=R and C. As most of the work in these two cases was completed, the study of the representation theory over other fields, like local fields and finite fields, began. The representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space, like the restricted root systems with Weyl groups, etc. Thus it is essential to study first this structure and the related geometry. In this paper we give a characterization of the isomorphy classes of these symmetric k-varieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers, p-adic numbers, finite fields and number fields.}, number={1}, journal={ADVANCES IN MATHEMATICS}, author={Helminck, AG}, year={2000}, month={Jul}, pages={1–117} }
@article{helminck_hilgert_neumam_olafsson_1999, title={A conjugacy theorem for symmetric spaces}, volume={313}, ISSN={["0025-5831"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0033412915&partnerID=MN8TOARS}, DOI={10.1007/s002080050282}, number={4}, journal={MATHEMATISCHE ANNALEN}, author={Helminck, AG and Hilgert, J and Neumam, A and Olafsson, G}, year={1999}, month={Apr}, pages={785–791} }
@article{helminck_helminck_1998, title={A class of parabolic k-subgroups associated with symmetric k-varieties}, volume={350}, ISSN={["1088-6850"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-33646875562&partnerID=MN8TOARS}, DOI={10.1090/S0002-9947-98-02029-7}, abstractNote={Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, σ an involution of G defined over k, H a k-open subgroup of the fixed point group of σ, Gk (resp. Hk) the set of k-rational points of G (resp. H) and Gk/Hk the corresponding symmetric k-variety. A representation induced from a parabolic k-subgroup of G generically contributes to the Plancherel decomposition of L2(Gk/Hk) if and only if the parabolic k-subgroup is σ-split. So for a study of these induced representations a detailed description of the Hk-conjucagy classes of these σ-split parabolic k-subgroups is needed.
In this paper we give a description of these conjugacy classes for general symmetric kvarieties. This description can be refined to give a more detailed description in a number of cases. These results are of importance for studying representations for real and p�-adic symmetric k-varieties.}, number={11}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Helminck, AG and Helminck, GF}, year={1998}, month={Nov}, pages={4669–4691} }
@article{helminck_1997, title={Tori invariant under an involutorial automorphism .2.}, volume={131}, ISSN={["0001-8708"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0031572783&partnerID=MN8TOARS}, DOI={10.1006/aima.1997.1633}, abstractNote={The geometry of the orbits of a minimal parabolick-subgroup acting on a symmetrick-variety is essential in several areas, but its main importance is in the study of the representations associated with these symmetrick-varieties (see for example [5, 6, 20, and 31]). Up to an action of the restricted Weyl group ofG, these orbits can be characterized by theHk-conjugacy classes of maximalk-split tori, which are stable underk-involutionθassociated with the symmetrick-variety. HereHis a openk-subgroup of the fixed point group ofθ. This is the second in a series of papers in which we characterize and classify theHk-conjugacy classes of maximalk-split tori. The first paper in this series dealt with the case of algebraically closed fields. In this paper we lay the foundation for a characterization and classification for the case of nonalgebraically closed fields. This includes a partial classification in the cases, where the base field is the real numbers, p-adic numbers, finite fields, and number fields.}, number={1}, journal={ADVANCES IN MATHEMATICS}, author={Helminck, AG}, year={1997}, month={Oct}, pages={1–92} }
@article{helminck_1996, title={Computing B-orbits on G/H}, volume={21}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0030075466&partnerID=MN8TOARS}, DOI={10.1006/jsco.1996.0008}, abstractNote={The orbits of a Borel subgroup acting on a symmetric varietyG/Hoccur in several areas of mathematics. For example, these orbits and their closures are essential in the study of HarishChandra modules (see Vogan, 1983). There are several descriptions of these orbits, but in practice it is actually very difficult and cumbersome to compute the orbits and their closures. Since the characterizations of theseorbits are very combinatorial in nature, this work could conceivably be done by a computer. In this paper we prove a number of additional properties of these orbits and combine these with properties of the various descriptions of these orbits to obtain an efficient algorithm. This algorithm can be implemented on a computer by using existing symbolic manipulation programs or by writing an independent program.}, number={2}, journal={Journal of Symbolic Computation}, author={Helminck, A.G.}, year={1996}, pages={169–209} }
@article{helminck_helminck_1995, title={Infinite-dimensional flag manifolds in integrable systems}, volume={41}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-34249759607&partnerID=MN8TOARS}, DOI={10.1007/BF00996107}, number={1-3}, journal={Acta Applicandae Mathematicae}, author={Helminck, G.F. and Helminck, A.G.}, year={1995}, pages={99–121} }
@inbook{helminck_helminck_1994, title={Holomorphic line bundles over Hilbert flag varieties}, volume={56}, booktitle={Algebraic groups and their generalizations: quantum and
infinite-dimensional methods (University Park, PA,
1991)}, publisher={Amer. Math. Soc., Providence, RI}, author={Helminck, A. G. and Helminck, G. F.}, year={1994}, pages={349–375} }
@inbook{helminck_1994, title={Symmetric $k$-varieties}, volume={56}, booktitle={Algebraic groups and their generalizations: classical methods
(University Park, PA, 1991)}, publisher={Amer. Math. Soc., Providence, RI}, author={Helminck, A. G.}, year={1994}, pages={233–279} }
@article{helminck_helminck_1994, title={The Structure of Hilbert Flag Varieties Dedicated to the memory of our father}, volume={30}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85024241886&partnerID=MN8TOARS}, DOI={10.2977/prims/1195165905}, abstractNote={In this paper we present a geometric realization of infinite dimensional analogues of the finite dimensional representations of the general linear group. This requires a detailed analysis of the structure of the flag varieties involved and the line bundles over them. In general the action of the restricted linear group can not be lifted to the line bundles and thus leads to central extensions of this group. It is determined exactly when these extensions are non-trivial. These representations are of importance in quantum field theory and in the framework of integrable systems. As an application, it is shown how the flag varieties occur in the latter context.}, number={3}, journal={Publications of the Research Institute for Mathematical Sciences}, author={Helminck, G.F. and Helminck, A.G.}, year={1994}, pages={401–441} }
@article{helminck_wang_1993, title={On rationality properties of involutions of reductive groups}, volume={99}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0002214015&partnerID=MN8TOARS}, DOI={10.1006/aima.1993.1019}, number={1}, journal={Advances in Mathematics}, author={Helminck, A.G. and Wang, S.P.}, year={1993}, pages={26–96} }
@inproceedings{helminck_1991, title={On groups with a Cartan involution}, booktitle={Proceedings of the Hyderabad Conference on Algebraic
Groups (Hyderabad, 1989)}, publisher={Manoj Prakashan, Madras}, author={Helminck, A. G.}, year={1991}, pages={151–192} }
@article{helminck_1991, title={Tori invariant under an involutorial automorphism, I}, volume={85}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0001618397&partnerID=MN8TOARS}, DOI={10.1016/0001-8708(91)90048-C}, abstractNote={Abstract The geometry of the orbits of a minimal parabolick-subgroup acting on a symmetrick-variety is essential in several areas, but its main importance is in the study of the representations associated with these symmetrick-varieties (see for example [5, 6, 20, and 31]). Up to an action of the restricted Weyl group ofG, these orbits can be characterized by theHk-conjugacy classes of maximalk-split tori, which are stable underk-involutionθassociated with the symmetrick-variety. HereHis a openk-subgroup of the fixed point group ofθ. This is the second in a series of papers in which we characterize and classify theHk-conjugacy classes of maximalk-split tori. The first paper in this series dealt with the case of algebraically closed fields. In this paper we lay the foundation for a characterization and classification for the case of nonalgebraically closed fields. This includes a partial classification in the cases, where the base field is the real numbers, p -adic numbers, finite fields, and number fields.}, number={1}, journal={Advances in Mathematics}, author={Helminck, A.G.}, year={1991}, pages={1–38} }
@inproceedings{helminck_1989, title={On the orbits of symmetric spaces under the action of parabolic subgroups}, url={http://dx.doi.org/10.1090/conm/088/999998}, DOI={10.1090/conm/088/999998}, booktitle={Invariant Theory}, publisher={American Mathematical Society}, author={Helminck, A. G.}, year={1989}, pages={435–447} }
@article{helminck_1988, title={Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces}, volume={71}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-38249028931&partnerID=MN8TOARS}, DOI={10.1016/0001-8708(88)90066-7}, abstractNote={Let G be a connected reductive algebraic group defined over an algebraically closed field F of characteristic not 2. Denote the Lie algebra of G by 9. In this paper we shall classify the isomorphism classes of ordered pairs of commuting involutorial automorphisms of G. This is shown to be independent of the characteristic of F and can be applied to describe all semisimple locally symmetric spaces together with their line structure. Involutorial automorphisms of g occur in several places in the literature. Cartan has already shown that for F= C, the isomorphism classes of involutorial automorphisms of g correspond bijectively to the isomorphism classes of real semisimple Lie algebras, which correspond in their turn to the isomorphism classes of Riemannian symmetric spaces (see Helgason [ 111). If one lifts this involution to the group G, then the present work gives a characteristic free description of these isomorphism classes. In a similar manner we can show that semisimple locally symmetric spaces correspond to pairs of commuting involutorial automorphisms of g. Namely let (go, a) be a semisimple locally symmetric pair; i.e., go is a real semisimple Lie algebra and rr E Aut(g,) an involution. Then by a result of Berger [2], there exists a Cartan involution 8 of go, such that 00 = ea. If we denote the complexilication of go by g, then o and 8 induce a pair of commuting involutions of g. Conversely, if c, 8 E Aut(g) are commuting involutions, then c and 8 determine two locally semisimple symmetric pairs. For if u is a O- and O-stable compact real form with conjugation r, then (ger, (~1 ge,) and (g,,, f3 I ger) are semisimple locally symmetric pairs where}, number={1}, journal={Advances in Mathematics}, author={Helminck, A.G.}, year={1988}, pages={21–91} }
@article{cohen_helminck_1988, title={Trilinear alternating forms on a vector space of dimension 7}, volume={16}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0037720879&partnerID=MN8TOARS}, DOI={10.1080/00927878808823558}, abstractNote={Abstract For vector spaces of dimension at most 7 over fields of cohomo-ogical dimension at most 1 (including algebraically closed fields and inite fields) all trilinear alternating forms and their isotropy groups are determined.}, number={1}, journal={Communications in Algebra}, author={Cohen, A.M. and Helminck, A.G.}, year={1988}, pages={1–25} }
@inbook{helminck_1986, title={A classification of semisimple symmetric pairs and their
restricted root system}, volume={5}, booktitle={Lie algebras and related topics (Windsor, Ont., 1984)}, publisher={Amer. Math. Soc., Providence, RI}, author={Helminck, A. G.}, year={1986}, pages={333–340} }
@book{koornwinder_hoogenboom_cohen_vries_helminck_1982, title={The structure of real semisimple Lie groups}, volume={49}, publisher={Mathematisch Centrum, Amsterdam}, author={Koornwinder, T. H. and Hoogenboom, B. and Cohen, A. M. and Vries, J. and Helminck, A. G.}, year={1982}, pages={v+141} }
@article{buell_helminck_klima_schaefer_wright_ziliak_2017, title={On the structure of generalized symmetric spaces of SLnFq)}, volume={45}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85028521780&partnerID=MN8TOARS}, DOI={10.1080/00927872.2017.1296458}, number={12}, journal={Communications in Algebra}, author={Buell, C. and Helminck, A. and Klima, V. and Schaefer, J. and Wright, C. and Ziliak, E.}, year={2017}, pages={5123–5136} }