{"id":40,"date":"2013-03-06T23:31:39","date_gmt":"2013-03-06T22:31:39","guid":{"rendered":"http:\/\/193.205.128.45\/marietti\/?page_id=40"},"modified":"2025-10-26T16:19:01","modified_gmt":"2025-10-26T15:19:01","slug":"publication","status":"publish","type":"page","link":"https:\/\/math-diism.univpm.it\/marietti\/publication\/","title":{"rendered":"Publications"},"content":{"rendered":"<h1>Publications of Mario\u00a0Marietti<\/h1>\n<h3>Reprints will be mailed upon request.<\/h3>\n<div class=\"elements-module__IeohX\"><\/div>\n<ul>\n<li>\n<h4 class=\"Typography-module__lVnit Typography-module__Cv8mo Typography-module__mZVLC Typography-module__ETlt8\"><a href=\"https:\/\/rdcu.be\/eMh8J\"><strong>Flipclasses and Combinatorial Invariance for <\/strong><\/a><span class=\"Typography-module__lVnit Typography-module__Nfgvc Button-module__Imdmt\"><strong>Kazhdan\u2013Lusztig polynomials<\/strong> (with F. Esposito)<br \/>\nSelecta Mathematica, 31, 98 2025 https:\/\/doi.org\/10.1007\/s00029-025-01099-6<\/span><\/h4>\n<\/li>\n<li>\n<div class=\"elements-module__IeohX\">\n<h4 class=\"Typography-module__lVnit Typography-module__Cv8mo Typography-module__mZVLC Typography-module__ETlt8\"><span class=\"Typography-module__lVnit Typography-module__Nfgvc Button-module__Imdmt\"><strong>A note on combinatorial invariance of Kazhdan\u2013Lusztig polynomials<\/strong> <\/span>(with F. <span class=\"Authors-module__umR1O\"><span class=\"Typography-module__lVnit Typography-module__Nfgvc Button-module__Imdmt\">Esposito and with an appendix\u00a0by Grant T. Barkley and Christian Gaetz)<\/span><\/span><\/h4>\n<\/div>\n<p><span class=\"Typography-module__lVnit Typography-module__fRnrd Typography-module__Nfgvc\">Bulletin of the London Mathematical Society, 2025<\/span><\/li>\n<li><a href=\"https:\/\/rdcu.be\/d4Eqw\"><strong>Kazhdan\u2013Lusztig R-polynomials, combinatorial invariance, and hypercube decompositions<\/strong><\/a> (with F. Brenti)<br \/>\n<i><\/i>Mathematische Zeitschrift, 309, 25 (2025). https:\/\/doi.org\/10.1007\/s00209-024-03632-3<\/li>\n<li><a href=\"https:\/\/academic.oup.com\/imrn\/advance-article-abstract\/doi\/10.1093\/imrn\/rnad063\/7098280?redirectedFrom=fulltext\"><strong>Motzkin combinatorics in linear degenerations of the flag variety<\/strong><\/a> (with G. Cerulli-Irelli and F. Esposito)<br \/>\nInternational Mathematics Research Notices, Vol. 2023, No. 22, pp. 19184\u201319204<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/abs\/pii\/S0021869322004665\">Bruhat intervals and parabolic cosets in arbitrary Coxeter groups<\/a><\/strong><br \/>\nJournal of Algebra, 614 (2023), 1-4<\/li>\n<li><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/abs\/pii\/S0021869321004051\"><strong>Root systems, affine subspaces, and projections<\/strong><\/a> (with P. Cellini)<br \/>\nJournal of Algebra, 587 (2021), 310-335<\/li>\n<li><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/abs\/pii\/S002186932030449X\"><strong>Pircon kernels and up-down symmetry<\/strong><\/a> (with F. Caselli)<br \/>\nJournal of Algebra, 565 (2021), 324-352<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/1904.10208\"><strong>Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids<\/strong><\/a><strong>\u00a0<\/strong>(with F. Caselli and M. D&#8217;Adderio)<br \/>\n<a href=\"https:\/\/academic.oup.com\/imrn\/article-abstract\/doi\/10.1093\/imrn\/rnaa124\/5856798?redirectedFrom=fulltext\">International Mathematics Research Notices<\/a>, Volume 2021, Issue 3, (2021), 1678\u20131698,<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0021869319303357\">Kazhdan\u2013Lusztig R-polynomials for pircons<\/a>, <\/strong><br \/>\nJournal of Algebra, 534 (2019), 245-272<\/li>\n<li><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0001870818302561\"><strong>The combinatorial invariance conjecture for parabolic Kazhdan&#8211;Lusztig polynomials of lower intervals<\/strong><\/a>,<br \/>\nAdvances in Mathematics, 335 (2018), 180-210<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0196885818300915\">Fixed points and adjacents for classical complex reflection groups<\/a>,<\/strong> (with\u00a0<a href=\"http:\/\/axp.mat.uniroma2.it\/~brenti\/brenti2.htm\">F.\u00a0Brenti<\/a>),<br \/>\nAdvances in Applied Mathematics, 101 (2018), 168-181<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0012365X17304211\" target=\"_blank\" rel=\"noopener noreferrer\">A simple characterization of special matchings in lower Bruhat intervals\u00a0<\/a><\/strong>(with <a href=\"https:\/\/www.unibo.it\/sitoweb\/fabrizio.caselli\/\">F. Caselli<\/a>)<strong>,<\/strong><br \/>\nDiscrete Mathematics<i>,\u00a0 <\/i>341 (2018), 851-862<\/p>\n<pre>We give a simple characterization of special matchings in lower Bruhat intervals (that is, intervals starting from the identity element) of Coxeter group. As a byproduct, we obtain some results on the action of special matchings.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0195669816301019\" target=\"_blank\" rel=\"noopener noreferrer\">Special matchings in Coxeter groups<\/a><\/strong> (with <a href=\"https:\/\/www.unibo.it\/sitoweb\/fabrizio.caselli\/\">F. Caselli<\/a>),<br \/>\nEuropean Journal of\u00a0Combinatorics<i>,\u00a0 61<\/i> (2017), 151-166<\/p>\n<pre>Special matchings are purely combinatorial objects associated with a partially ordered set, which have applications in Coxeter group theory. We provide an explicit characterization and a complete classification of all special matchings of any lower Bruhat interval. The results hold in any arbitrary Coxeter group and have also applications in the study of the corresponding parabolic Kazhdan--Lusztig polynomials.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/www.ams.org\/journals\/tran\/2016-368-07\/S0002-9947-2015-06676-8\/S0002-9947-2015-06676-8.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">Special matchings and parabolic Kazhdan-Lusztig polynomials,<br \/>\n<\/a><\/strong>Transactions of the American Mathematical Society, 368 (2016), 5247-5269<\/p>\n<pre>We prove that the combinatorial concept of a special matching can be used to compute the parabolic Kazhdan--Lusztig polynomials of doubly laced Coxeter groups and of dihedral Coxeter groups. In particular, for this class of groups which includes all Weyl groups, our results generalize to the parabolic setting the main results in [Advances in Math. {202} (2006), 555-601]. As a consequence, the parabolic Kazhdan--Lusztig polynomial indexed by $u$ and $v$ depends only on the poset structure of the Bruhat interval from the identity element to $v$ and on which elements of that interval are minimal coset representatives.<\/pre>\n<\/li>\n<li><strong><a href=\"http:\/\/www.mat.univie.ac.at\/~slc\/\" target=\"_blank\" rel=\"noopener noreferrer\">Polar root polytopes that are zonotopes<\/a> <\/strong>(with <strong><a href=\"http:\/\/paolacellini.unich.it\/\">P. Cellini<\/a>)<\/strong>,<br \/>\n<i><a href=\"http:\/\/www.mat.univie.ac.at\/~slc\/\" target=\"_blank\" rel=\"noopener noreferrer\">S\u00e9minaire Lotharingien de Combinatoire<\/a>,<\/i>\u00a073 (2015), B73a<\/p>\n<pre>Let $\\pol_{\\Phi}$ be the root polytope of a finite irreducible crystallographic root system $\\Phi$, i.e., the convex hull of all roots in $\\Phi$. The polar of $\\pol_{\\Phi}$, denoted $\\pol_{\\Phi}^*$, coincides with the union of the orbit of the fundamental alcove under the action of the Weyl group. In this paper, we establishes which polytopes $\\pol_{\\Phi}^*$ are zonotopes and which are not. The proof is constructive.<\/pre>\n<\/li>\n<li><strong><strong><strong><a href=\"https:\/\/academic.oup.com\/imrn\/article-abstract\/2015\/12\/4392\/677744\" target=\"_blank\" rel=\"noopener noreferrer\">Root polytopes and Borel subalgebras<\/a><\/strong><\/strong><\/strong> (with <a href=\"http:\/\/paolacellini.unich.it\/\">P. Cellini<\/a>),<strong><strong><br \/>\n<\/strong><\/strong><i><a href=\"http:\/\/imrn.oxfordjournals.org\/content\/by\/year\" target=\"_blank\" rel=\"noopener noreferrer\">International Mathematics Research Notices<\/a>, <\/i><span class=\"slug-pub-date\">12, \u00a02015 (2015),<\/span><span class=\"slug-issue\">\u00a0<\/span><span class=\"slug-pages\">4392-4420<\/span><strong><strong><i><br \/>\n<\/i><\/strong><\/strong><a href=\"http:\/\/imrn.oxfordjournals.org\/cgi\/content\/abstract\/rnu070?%20ijkey=BEaSeiI38gUzwzM&amp;keytype=ref\" target=\"_blank\" rel=\"noopener noreferrer\">doi:\u00a0<span class=\"slug-doi\" title=\"10.1093\/imrn\/rnu070\">10.1093\/imrn\/rnu070<\/span><\/a><\/p>\n<pre>Let $\\Phi$ be a finite crystallographic irreducible root system and $\\mathcal P_{\\Phi}$ be the convex hull of the roots in $\\Phi$. We give a uniform explicit description of the polytope $\\mathcal P_{\\Phi}$, analyze the algebraic-combinatorial structure of its faces, and provide connections with the Borel subalgebra of the associated Lie algebra. We also give several enumerative results.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/link.springer.com\/article\/10.1007\/s10801-013-0458-5\">Root polytopes and abelian ideals<\/a><\/strong> (with <a href=\"http:\/\/paolacellini.unich.it\/\">P. Cellini<\/a>),<br \/>\n<i><a href=\"http:\/\/www.springerlink.com\/content\/1572-9192\/\">Journal of Algebraic Combinatorics<\/a>, <\/i>39 (2014), no 3, 607-645<br \/>\n<a href=\"http:\/\/link.springer.com\/article\/10.1007%2Fs10801-013-0458-5\" target=\"_blank\" rel=\"noopener noreferrer\">doi: 10.1007\/s10801-013-0458-5<\/a>\u00a0\u00a0<b>\u00a0 <\/b><\/p>\n<pre>We study the root polytope $\\mathcal P_\\Phi$ of a finite irreducible crystallographic root system $\\Phi$ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system $\\Phi$. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of $\\mathcal P_\\Phi$ and analyze its relation with the facets of $\\mathcal P_\\Phi$. For $\\Phi$ of type $A_n$ or $C_n$, we show that the orbits of some special subsets of abelian ideals under the action of the Weyl group parametrize a triangulation of $\\mathcal P_\\Phi$. We show that this triangulation restricts to a triangulation of the positive root polytope $\\mathcal P_\\Phi^+$.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0195669812000601\">Conical and Spherical graphs<\/a>\u00a0<\/strong>(with <a href=\"http:\/\/homepages.warwick.ac.uk\/~maskal\/zone\">D.\u00a0Testa<\/a>),<br \/>\n<i><a href=\"http:\/\/www.elsevier.com\/wps\/find\/journaldescription.cws_home\/622824\/description#description\">European Journal of\u00a0Combinatorics<\/a>,\u00a0<\/i>33 (2012), 1606-1618\u00a0\u00a0<b>\u00a0\u00a0<\/b><\/p>\n<pre>We introduce and study the notions of conical and spherical graphs.\u00a0\u00a0We show that these mutually exclusive properties, which have a geometric interpretation, provide links between apparently unrelated classical concepts such as dominating sets, independent dominating sets, edge covers, and the\u00a0homotopy\u00a0type of an associated\u00a0simplicial\u00a0complex. In particular, we solve the problem of characterizing the forests whose dominating sets of minimum cardinality are also independent. To establish these connections, we prove a formula to compute the Euler characteristic of an arbitrarysimplicial\u00a0complex from a set of generators of its Stanley-Reisner\u00a0ideal.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0196885811000078\">Kazhdan-Lusztig\u00a0polynomials, tight quotients and\u00a0Dyck\u00a0superpartitions<\/a>\u00a0<\/strong>(with\u00a0<a href=\"http:\/\/axp.mat.uniroma2.it\/~brenti\/brenti2.htm\">F.\u00a0Brenti\u00a0<\/a>and\u00a0<a href=\"http:\/\/www.mat.uniroma1.it\/~incitti\/\">F.\u00a0Incitti<\/a>),<br \/>\n<i><a href=\"http:\/\/www.elsevier.com\/wps\/find\/journaldescription.cws_home\/622776\/description?navopenmenu=-2\">Advances in Applied Mathematics<\/a><\/i>, 47 (2011), 589-614<\/p>\n<pre>We give an explicit closed combinatorial formula for the parabolic\u00a0Kazhdan-Lusztig\u00a0polynomials of the tight quotients of the symmetric group. This involves a combinatorial concept which does not seem to have been studied before. Our main result implies that these polynomials are always either zero or a\u00a0monic\u00a0power of $q$, as well as the main result of [Pacific J. Math., 207 (2002), 257-286] on the parabolic\u00a0Kazhdan-Lusztig\u00a0polynomials of the maximal quotients of the symmetric group.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0195669809001334\">Parabolic\u00a0Kazhdan-Lusztig\u00a0and $R$-Polynomials for Boolean Elements in the symmetric group<\/a>,<br \/>\n<\/strong><a href=\"http:\/\/www.elsevier.com\/wps\/find\/journaldescription.cws_home\/622824\/description#description\"><i>European Journal of\u00a0Combinatorics<\/i><\/a>, 31 (2010), 908-92<\/p>\n<pre>The parabolic analogue of the\u00a0Kazhdan-Lusztig\u00a0and $R$-polynomials has been introduced by\u00a0Deodhar\u00a0[J. Algebra, 111 (1987), 483-506]. Answering a question of\u00a0Brenti, we give closed combinatorial product formulae for the parabolic $R$-polynomials and for the parabolic\u00a0Kazhdan-Lusztig\u00a0polynomials of type $q$ in the case that the indexing permutations are smaller than the top transposition in\u00a0Bruhat\u00a0order. These formulae are valid in complete generality on the parabolic subgroup $W_J$.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/epubs.siam.org\/doi\/10.1137\/070695034\">A combinatorial characterization of\u00a0Coxeter\u00a0groups<\/a>,<br \/>\n<\/strong><em><a href=\"http:\/\/www.siam.org\/journals\/sidma.php\">SIAM Journal on Discrete Mathematics<\/a><\/em>, 23 (2009), no 1, 319-332<\/p>\n<pre>In this paper we give a purely combinatorial characterization of\u00a0Coxeter\u00a0groups in terms of their partial order structure under\u00a0Bruhat\u00a0order. The result is based on the recently introduced concept of special matching and is achieved by proving an analogue of Tits' Word Theorem. As a consequence of the proof of our main result, we obtain a result about\u00a0shellability.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/emis.u-strasbg.fr\/journals\/EJC\/Volume_15\/PDF\/v15i1r101.pdf\">A uniform approach to complexes arising from forests<\/a>\u00a0<\/strong>(with\u00a0<a href=\"http:\/\/homepages.warwick.ac.uk\/~maskal\/zone\">D.Testa<\/a>),<br \/>\n<em><a href=\"http:\/\/www.combinatorics.org\/\">The Electronic Journal of Combinatorics<\/a><\/em>, 15 (2008), no 1, Research Paper 101, 18 pp.<\/p>\n<pre>In this paper we present a unifying approach to study the\u00a0homotopy\u00a0type of several complexes arising from forests. We show that this method applies uniformly to many complexes that have been extensively studied in the recent years.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/link.springer.com\/article\/10.1007\/s00454-008-9081-y\">Cores of\u00a0simplicial\u00a0complexes<\/a>\u00a0<\/strong>(with\u00a0<a href=\"http:\/\/homepages.warwick.ac.uk\/~maskal\/zone\">D.Testa<\/a>),<br \/>\n<i><a href=\"http:\/\/www.springer.com\/math\/numbers\/journal\/454\">Discrete and Computational Geometry<\/a>,<\/i>\u00a040 (2008), no 3, 444-468<\/p>\n<pre>We introduce a method to reduce the study of the topology of a\u00a0simplicial\u00a0complex to that of a simpler one. Applying this method to complexes arising from graphs, we give topological meaning to classical graph invariants. As a consequence, we answer some questions raised in [European J.~Combin.~27 (2006), no.~6, 906-923] on the independence complex and the dominance complex of a forest and we obtain improved algorithms to compute their\u00a0homotopy\u00a0type.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0195669807002247\">On a duality in\u00a0Coxeter\u00a0groups<\/a>,<br \/>\n<\/strong><i><a href=\"http:\/\/www.elsevier.com\/wps\/find\/journaldescription.cws_home\/622824\/description#description\">European Journal of\u00a0Combinatorics<\/a>,\u00a0\u00a0<\/i>29 (2008), 1555-1562<\/p>\n<pre>In [J. Integer Seq. 8 (2005), no. 3, Article 05.3.8.] Stanley gives certain enumerative identities revealing a duality between descent sets and connectivity sets of the symmetric group. In this paper we generalize these identities to all\u00a0Coxeter\u00a0groups. The proofs are obtained by giving theseidentites\u00a0an algebraic\u00a0explaination\u00a0in terms of parabolic subgroups,\u00a0coset\u00a0representatives, and\u00a0Poincar\\'e series, and by a formal argument in terms of inclusion-exclusion-like matrices.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/link.springer.com\/article\/10.1007\/s00013-007-2207-2\">Special\u00a0matchings\u00a0and\u00a0Coxeter\u00a0groups<\/a>\u00a0<\/strong>(with\u00a0<a href=\"http:\/\/axp.mat.uniroma2.it\/~brenti\/brenti2.htm\">F.\u00a0Brenti\u00a0<\/a>and\u00a0<a href=\"https:\/\/www.unibo.it\/sitoweb\/fabrizio.caselli\/\">F.\u00a0Caselli<\/a>),<br \/>\n<i><a href=\"http:\/\/www.springerlink.com\/content\/1420-8938\/\">Archiv\u00a0der\u00a0Mathematik<\/a>,<\/i>\u00a089 (2007), no 4, 298-310<b><\/b><\/p>\n<pre>In the recent paper [Adv. Applied Math., 38 (2007), 210--226] it is proved that the special\u00a0matchings\u00a0of permutations generate a\u00a0Coxeter\u00a0group. In this paper we generalize this result to a class of\u00a0Coxeter\u00a0groups which includes many\u00a0Weyl\u00a0and affine\u00a0Weyl\u00a0groups. Our proofs are simpler, and shorter, than those in {[}loc. cit.]<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/link.springer.com\/article\/10.1007\/s10801-007-0061-8\">Algebraic and combinatorial properties of zircons<\/a>,<br \/>\n<\/strong><i><a href=\"http:\/\/www.springerlink.com\/content\/1572-9192\/\">Journal of Algebraic\u00a0Combinatorics<\/a>,<\/i>\u00a026\u00a0(2007), no 3, 363-382<\/p>\n<pre>In this paper we introduce and study a new class of\u00a0posets, that we call zircons, which includes all\u00a0Coxeter\u00a0groups partially ordered by\u00a0Bruhat\u00a0order. We prove that many of the properties of\u00a0Coxeter\u00a0groups extend to zircons often with simpler proofs: in particular, zircons are\u00a0Eulerian\u00a0posets\u00a0and the\u00a0Kazhdan-Lusztig\u00a0construction of the\u00a0Kazhdan-Lusztig\u00a0representations can be carried out in the context of zircons. Moreover, for any zircon $Z$, we construct and count all balanced and exact\u00a0labelings\u00a0(used in the construction of the Bernstein-Gelfand-Gelfand\u00a0resolutions in the case that $Z$ is the\u00a0Weyl\u00a0group of a\u00a0Kac-Moody algebra).<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0196885806000996\">Special\u00a0matchings\u00a0and permutations in\u00a0Bruhat\u00a0orders<\/a>\u00a0<\/strong>(with\u00a0<a href=\"http:\/\/axp.mat.uniroma2.it\/~brenti\/brenti2.htm\">F.\u00a0Brenti\u00a0<\/a>and\u00a0<a href=\"https:\/\/www.unibo.it\/sitoweb\/fabrizio.caselli\/\">F.\u00a0Caselli<\/a>),<br \/>\n<i><a href=\"http:\/\/www.elsevier.com\/wps\/find\/journaldescription.cws_home\/622776\/description?navopenmenu=-2\">Advances in Applied Mathematics<\/a><\/i>, 38 (2007), 210-226<b><\/b><\/p>\n<pre>In this paper we show that, for any permutation v, the special\u00a0matchings\u00a0of the lower\u00a0Bruhat\u00a0interval [e,v], considered as involutions on [e,v], generate a\u00a0Coxeter\u00a0system. This gives new necessary conditions on an abstract\u00a0poset\u00a0to be isomorphic to a lower\u00a0Bruhat\u00a0interval of the symmetric group, and also answers in the affirmative, in the symmetric group case, a problem posed in [F.\u00a0Brenti, F.\u00a0Caselli, M.\u00a0Marietti, Advances in Math., 202 (2006), 555-601].<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/academic.oup.com\/imrn\/article-abstract\/doi\/10.1155\/IMRN\/2006\/29407\/661984?redirectedFrom=fulltext\">Diamonds and\u00a0Hecke\u00a0algebra representations<\/a>\u00a0<\/strong>(with\u00a0<a href=\"http:\/\/axp.mat.uniroma2.it\/~brenti\/brenti2.htm\">F.\u00a0Brenti\u00a0<\/a>and\u00a0<a href=\"https:\/\/www.unibo.it\/sitoweb\/fabrizio.caselli\/\">F.\u00a0Caselli<\/a>),<br \/>\n<i><a href=\"http:\/\/imrn.oxfordjournals.org\/content\/by\/year\" target=\"_blank\" rel=\"noopener noreferrer\">In<\/a><a href=\"http:\/\/imrn.oxfordjournals.org\/content\/by\/year\" target=\"_blank\" rel=\"noopener noreferrer\">ternational Mathematics Research Notice<\/a><a href=\"http:\/\/imrn.oxfordjournals.org\/content\/by\/year\" target=\"_blank\" rel=\"noopener noreferrer\">s<\/a><\/i>, vol. 2006, Article ID 29407, 34 pages, 2006<\/p>\n<pre>In this work we show that\u00a0Kazhdan\u00a0and\u00a0Lusztig's\u00a0construction of\u00a0Hecke\u00a0algebra representations introduced in [D.\u00a0Kazhdan, G.\u00a0Lusztig, Invent. Math., 53 (1979), 165-184] can be carried out in a much more general (and entirely combinatorial) context. More precisely, we introduce a new class of partially ordered sets, that we call diamonds, which have a very rich combinatorial and algebraic structure and which include all\u00a0Coxetergroups partially ordered by\u00a0Bruhat\u00a0order. We prove that one can define a family of polynomials indexed by pairs of elements in the diamond which reduce to the\u00a0Kazhdan-Lusztig\u00a0polynomials in the case that the diamond is a\u00a0Coxeter\u00a0group. We then show that every diamond contains in a natural way a\u00a0Coxeter\u00a0group and hence a\u00a0Hecke\u00a0algebra. Finally we show that this\u00a0Coxeter\u00a0group and the corresponding\u00a0Hecke\u00a0algebra act naturally on the diamond, and that the resulting representations include those constructed by\u00a0Kazhdan\u00a0and\u00a0Lusztig, but contain several new ones.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0001870805000939\">Special\u00a0matchings\u00a0and\u00a0Kazhdan-Lusztig\u00a0polynomials<\/a>\u00a0<\/strong>(with\u00a0<a href=\"http:\/\/axp.mat.uniroma2.it\/~brenti\/brenti2.htm\">F.\u00a0Brenti\u00a0<\/a>and\u00a0<a href=\"https:\/\/www.unibo.it\/sitoweb\/fabrizio.caselli\/\">F.\u00a0Caselli<\/a>),<br \/>\n<i><a href=\"http:\/\/www.elsevier.com\/wps\/find\/journaldescription.cws_home\/622779\/description#description\">Advances in Mathematics<\/a>, <\/i>202 (2006), 555-601<\/p>\n<pre>In this paper we show that the combinatorial concept of a special matching plays a fundamental role in the computation of the Kazhdan-Lusztig polynomials. Our results also imply, and generalize, the recent one in [F. Du\u00a0Cloux, Advances in Math., 180 (2003), 146-175] on the combinatorial invariance of\u00a0Kazhdan-Lusztig\u00a0polynomials.<\/pre>\n<\/li>\n<li><strong><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0012365X06000835\">Formulas for multi-parameter classes of\u00a0Kazhdan-Lusztig\u00a0polynomials in S(n)<\/a>\u00a0<\/strong>(with\u00a0<a href=\"https:\/\/www.unibo.it\/sitoweb\/fabrizio.caselli\/\">F.\u00a0Caselli<\/a>),<br \/>\n<em id=\"__mceDel\"><i><a href=\"http:\/\/www.elsevier.com\/wps\/find\/journaldescription.cws_home\/505610\/description#description\">Discrete Mathematics<\/a><\/i><\/em>, 306 (2006), 711-725<\/p>\n<pre>In this paper we give explicit formulas for several classes of\u00a0Kazhdan-Lusztig\u00a0polynomials of the symmetric group which are related to others already considered in the literature. In particular, we generalize two theorems of\u00a0\u00a0Brenti\u00a0and\u00a0Simion\u00a0 [F.\u00a0Brenti, R.\u00a0Simion, Explicit formulae for someKazhdan-Lusztig\u00a0polynomials, J. Algebraic\u00a0Combin. {11} (2000) 187--196], we prove an unpublished conjecture of\u00a0Brenti, and we treat the case missing in Theorem 4.8 of [F.\u00a0Caselli, {Proof of two conjectures of\u00a0Brenti\u00a0and\u00a0Simion\u00a0on\u00a0Kazhdan-Lusztig\u00a0polynomials}, J. Algebraic\u00a0Combin.\u00a018 (2003), 171--187].<\/pre>\n<\/li>\n<li><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0021869305005569\"><strong>Boolean elements in\u00a0Kazhdan-Lusztig\u00a0theory<\/strong><\/a>,<br \/>\n<i><a href=\"http:\/\/www.elsevier.com\/locate\/issn\/0021-8693\">Journal of\u00a0 Algebra<\/a><\/i>, 295 (2006), no 1, 1-26<\/p>\n<pre>Kazhdan-Lusztig\u00a0polynomials have been proven to play an important role in different fields. Despite this, there are still few explicit formulae for them. In this paper we give closed product formulae for the\u00a0Kazhdan-Lusztig\u00a0polynomials indexed by Boolean elements in a class of\u00a0Coxeter\u00a0systems that we call linear. Boolean elements are elements smaller than a reflection that admits a reduced expression of the form $s_{1}\\ldots\u00a0s_{n-1}s_{n}s_{n-1}\\ldots\u00a0s_{1}$ ($s_{i}\\in S$, $s_{i}\\neq\u00a0s_{j}$ if $i\\neq\u00a0j$). Then we provide several applications of this result concerning\u00a0 the combinatorial invariance of the\u00a0Kazhdan-Lusztig\u00a0polynomials, the classification of the pairs $(u,v)$ with $u\\prec\u00a0v$, the\u00a0Kazhdan-Lusztig\u00a0elements and the intersection homology\u00a0Poincar\\'e polynomials of the Schubert varieties.<\/pre>\n<\/li>\n<li><strong><a href=\"http:\/\/Closed product formulas for certain R-polynomials,\" data-wplink-url-error=\"true\">Closed product formulas for certain R-polynomials<\/a>,<br \/>\n<\/strong><i><a href=\"http:\/\/www.elsevier.com\/wps\/find\/journaldescription.cws_home\/622824\/description#description\">European Journal of\u00a0Combinatorics<\/a>, <\/i>23 (2002), 57-62<\/p>\n<pre>$R$-polynomials get their importance from the fact that they are used to define and compute the\u00a0Kazhdan-Lusztig\u00a0polynomials, which have applications in several fields. In this paper we give a closed product formula for certain $R$-polynomials valid for every\u00a0Coxeter\u00a0group. This result implies a conjecture due to F.\u00a0Brenti\u00a0about the symmetric groups.<\/pre>\n<table class=\"MsoNormalTable\" border=\"0\" width=\"547\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 700.5pt;padding: 1.5pt 1.5pt 1.5pt 1.5pt\" width=\"934\">\n<h2><span class=\"SpellE\"><span style=\"color: #993300\">\u00a0<\/span><\/span><\/h2>\n<h2>Proceedings and chapters in books<\/h2>\n<ul>\n<li><strong>Root polytopes of crystallographic root systems<\/strong>,<br \/>\nin\u00a0<em>Combinatorial Methods\u00a0in Topology and Algebra,<br \/>\n<\/em>B. Benedetti,E. Delucchi, L. Moci\u00a0Editors, 2015, Springer<\/li>\n<li><span class=\"SpellE\"><b><span lang=\"EN-US\">Kazhdan<\/span><\/b><\/span><b><span lang=\"EN-US\">\u2013<span class=\"SpellE\">Lusztig<\/span> polynomials, tight quotients and <span class=\"SpellE\">Dyck<\/span> <span class=\"SpellE\">superpartition<\/span>,<br \/>\n<\/span><\/b><span class=\"SpellE\"><span lang=\"EN-US\">Oberwolfach<\/span><\/span><span lang=\"EN-US\"><span lang=\"EN-US\"> Report<b> <\/b>No. 15\/2010<b>, <\/b>861-863<br \/>\n<\/span><\/span><\/li>\n<\/ul>\n<hr \/>\n<h2><\/h2>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 700.5pt;padding: 1.5pt 1.5pt 1.5pt 1.5pt\" width=\"934\">\n<h2><span class=\"SpellE\"><strong>Other<\/strong><\/span><strong> <span class=\"SpellE\">papers<\/span> and <span class=\"SpellE\">publications<\/span><\/strong><\/h2>\n<ul type=\"disc\">\n<li style=\"list-style-type: none\">\n<ul type=\"disc\">\n<li class=\"MsoNormal\"><b><span lang=\"EN-US\"><a href=\"http:\/\/dottorato.math.unipd.it\/sites\/default\/files\/SemDott1011_note.pdf\"><span lang=\"IT\">An <span class=\"SpellE\">introduction<\/span> to <span class=\"SpellE\">Coxeter<\/span> <span class=\"SpellE\">group<\/span> <span class=\"SpellE\">theory<\/span><\/span><\/a><\/span>,<br \/>\n<\/b>Seminario Dottorato 2010\/11, Scuole di Dottorato in Matematica Pura e Matematica Computazionale, Dipartimento di Matematica, Universit\u00e0 di Padova<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul type=\"disc\">\n<li style=\"list-style-type: none\">\n<ul type=\"disc\">\n<li class=\"MsoNormal\"><b><a href=\"https:\/\/www.garzanti.it\/libri\/aa-vv-matematica-9788811505259\/\">La <span class=\"SpellE\">Garzantina<\/span> di Matematica<\/a>,<br \/>\n<\/b>Edizioni Garzanti (collaboratore per le voci di topologia)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul type=\"disc\">\n<li style=\"list-style-type: none\">\n<ul type=\"disc\">\n<li class=\"MsoNormal\"><span class=\"SpellE\"><b><span lang=\"EN-US\"><strong><a href=\"https:\/\/www.mittag-leffler.se\/preprints\/0405s\/info.php?id=34\">Combinatorial properties of zircons<\/a>,<br \/>\n<\/strong><\/span><\/b><span lang=\"EN-US\">Institut Mittag-Leffler, The Royal Swedish Academy of Sciences<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul type=\"disc\">\n<li style=\"list-style-type: none\">\n<ul type=\"disc\">\n<li class=\"MsoNormal\"><b><span lang=\"EN-US\" style=\"color: windowtext\"><a href=\"https:\/\/www.mittag-leffler.se\/preprints\/0405s\/info.php?id=07\">Special <span class=\"SpellE\">matchings<\/span> and permutations<\/a> <\/span><\/b><span lang=\"EN-US\" style=\"color: windowtext\">(with <\/span><span style=\"color: windowtext\"><a href=\"http:\/\/axp.mat.uniroma2.it\/%7Ebrenti\/\"><span lang=\"EN-US\" style=\"color: windowtext\">F. <span class=\"SpellE\">Brenti<\/span> <\/span><\/a><\/span><span lang=\"EN-US\" style=\"color: windowtext\">and <\/span><span style=\"color: windowtext\"><a href=\"http:\/\/www.dm.unibo.it\/%7Ecaselli\"><span lang=\"EN-US\" style=\"color: windowtext\">F. <span class=\"SpellE\">Caselli<\/span><\/span><\/a><\/span><span style=\"color: windowtext\"><span lang=\"EN-US\">),<br \/>\n<span class=\"SpellE\">Institut<\/span> <span class=\"SpellE\">Mittag-Leffler<\/span>, The Royal Swedish Academy of Sciences<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul type=\"disc\">\n<li style=\"list-style-type: none\">\n<ul type=\"disc\">\n<li class=\"MsoNormal\"><b><span lang=\"EN-US\" style=\"color: windowtext\"><a href=\"http:\/\/www.mat.uniroma1.it\/%7Ecombinat\/quaderni\/quad10E\/10E_marietti.html\"><span class=\"SpellE\"><span lang=\"IT\">Kazhdan-Lusztig<\/span><\/span><span lang=\"IT\"> <span class=\"SpellE\">polynomials<\/span> for <span class=\"SpellE\">boolean<\/span> <span class=\"SpellE\">elements<\/span> in linear <span class=\"SpellE\">Coxeter<\/span> <span class=\"SpellE\">systems<\/span><\/span><\/a><\/span><\/b><b><span style=\"color: windowtext\">,<br \/>\n<\/span><\/b><span style=\"color: windowtext\"><span style=\"color: windowtext\">Quaderni Elettronici del Seminario di Geometria Combinatoria 10E (Maggio 2003), Universit\u00e0 degli Studi di Roma &#8220;La Sapienza&#8221; &#8211; Dipartimento di Matematica<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul type=\"disc\">\n<li class=\"MsoNormal\"><b><a href=\"http:\/\/193.205.128.45\/marietti\/wp-content\/uploads\/sites\/9\/2013\/04\/mariotesi1.pdf\">Kazhdan-Lusztig theory: Boolean elements, special matchings and combinatorial invariance,<\/a><\/b><br \/>\n<span class=\"SpellE\"><span style=\"color: windowtext\">Ph.D<\/span><\/span><span style=\"color: windowtext\">. <span class=\"SpellE\">Thesis<\/span>, Universit\u00e0 di Roma &#8220;La Sapienza&#8221;<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 700.5pt;padding: 1.5pt 1.5pt 1.5pt 1.5pt\" width=\"934\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 700.5pt;padding: 1.5pt 1.5pt 1.5pt 1.5pt\" width=\"934\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Publications of Mario\u00a0Marietti Reprints will be mailed upon request. Flipclasses and Combinatorial Invariance for Kazhdan\u2013Lusztig polynomials (with F. Esposito) Selecta Mathematica, 31, 98 2025 https:\/\/doi.org\/10.1007\/s00029-025-01099-6 A note on combinatorial invariance of Kazhdan\u2013Lusztig polynomials (with F. Esposito and with an appendix\u00a0by &hellip; <a href=\"https:\/\/math-diism.univpm.it\/marietti\/publication\/\">Continua a leggere<span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":8,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"open","template":"showcase.php","meta":{"footnotes":""},"class_list":["post-40","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math-diism.univpm.it\/marietti\/wp-json\/wp\/v2\/pages\/40","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math-diism.univpm.it\/marietti\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math-diism.univpm.it\/marietti\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math-diism.univpm.it\/marietti\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/math-diism.univpm.it\/marietti\/wp-json\/wp\/v2\/comments?post=40"}],"version-history":[{"count":142,"href":"https:\/\/math-diism.univpm.it\/marietti\/wp-json\/wp\/v2\/pages\/40\/revisions"}],"predecessor-version":[{"id":1628,"href":"https:\/\/math-diism.univpm.it\/marietti\/wp-json\/wp\/v2\/pages\/40\/revisions\/1628"}],"wp:attachment":[{"href":"https:\/\/math-diism.univpm.it\/marietti\/wp-json\/wp\/v2\/media?parent=40"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}