Publications

Publications of Mario Marietti

Reprints will be mailed upon request.

  • Motzkin combinatorics in linear degenerations of the flag variety (with G. Cerulli-Irelli and F. Esposito)
    International Mathematics Research Notices, Vol. 2023, No. 22, pp. 19184–19204
  • Bruhat intervals and parabolic cosets in arbitrary Coxeter groups,
    Journal of Algebra, 614 (2023), 1-4
  • Root systems, affine subspaces, and projections (with P. Cellini)
    Journal of Algebra, 587 (2021), 310-335
  • Pircon kernels and up-down symmetry (with F. Caselli)
    Journal of Algebra, 565 (2021), 324-352
  • Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids, (with F. Caselli and M. D’Adderio)
    International Mathematics Research Notices, Volume 2021, Issue 3, (2021), 1678–1698,
  • Kazhdan–Lusztig R-polynomials for pircons,
    Journal of Algebra, 534 (2019), 245-272
  • The combinatorial invariance conjecture for parabolic Kazhdan–Lusztig polynomials of lower intervals,
    Advances in Mathematics, 335 (2018), 180-210
  • Fixed points and adjacents for classical complex reflection groups, (with F. Brenti),
    Advances in Applied Mathematics, 101 (2018), 168-181
  • A simple characterization of special matchings in lower Bruhat intervals (with F. Caselli),
    Discrete Mathematics341 (2018), 851-862

    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.
  • Special matchings in Coxeter groups (with F. Caselli),
    European Journal of Combinatorics,  61 (2017), 151-166

    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.
  • Special matchings and parabolic Kazhdan-Lusztig polynomials,
    Transactions of the American Mathematical Society, 368 (2016), 5247-5269

    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.
  • Polar root polytopes that are zonotopes (with P. Cellini),
    Séminaire Lotharingien de Combinatoire, 73 (2015), B73a

    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.
  • Root polytopes and Borel subalgebras (with P. Cellini),
    International Mathematics Research Notices, 12,  2015 (2015), 4392-4420
    doi: 10.1093/imrn/rnu070

    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.
  • Root polytopes and abelian ideals (with P. Cellini),
    Journal of Algebraic Combinatorics, 39 (2014), no 3, 607-645
    doi: 10.1007/s10801-013-0458-5   

    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 $\mathcal P_\Phi^+$.
  • Conical and Spherical graphs (with D. Testa),
    European Journal of Combinatorics33 (2012), 1606-1618    

    We introduce and study the notions of conical and spherical graphs.  We 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 homotopy type of an associated simplicial complex. 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 complex from a set of generators of its Stanley-Reisner ideal.
  • Kazhdan-Lusztig polynomials, tight quotients and Dyck superpartitions (with F. Brenti and F. Incitti),
    Advances in Applied Mathematics, 47 (2011), 589-614

    We give an explicit closed combinatorial formula for the parabolic Kazhdan-Lusztig polynomials 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 monic power of $q$, as well as the main result of [Pacific J. Math., 207 (2002), 257-286] on the parabolic Kazhdan-Lusztig polynomials of the maximal quotients of the symmetric group.
  • Parabolic Kazhdan-Lusztig and $R$-Polynomials for Boolean Elements in the symmetric group,
    European Journal of Combinatorics, 31 (2010), 908-92

    The parabolic analogue of the Kazhdan-Lusztig and $R$-polynomials has been introduced by Deodhar [J. Algebra, 111 (1987), 483-506]. Answering a question of Brenti, we give closed combinatorial product formulae for the parabolic $R$-polynomials and for the parabolic Kazhdan-Lusztig polynomials of type $q$ in the case that the indexing permutations are smaller than the top transposition in Bruhat order. These formulae are valid in complete generality on the parabolic subgroup $W_J$.
  • A combinatorial characterization of Coxeter groups,
    SIAM Journal on Discrete Mathematics, 23 (2009), no 1, 319-332

    In this paper we give a purely combinatorial characterization of Coxeter groups in terms of their partial order structure under Bruhat order. 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 shellability.
  • A uniform approach to complexes arising from forests (with D.Testa),
    The Electronic Journal of Combinatorics, 15 (2008), no 1, Research Paper 101, 18 pp.

    In this paper we present a unifying approach to study the homotopy type 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.
  • Cores of simplicial complexes (with D.Testa),
    Discrete and Computational Geometry, 40 (2008), no 3, 444-468

    We introduce a method to reduce the study of the topology of a simplicial complex 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 homotopy type.
  • On a duality in Coxeter groups,
    European Journal of Combinatorics,  29 (2008), 1555-1562

    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 Coxeter groups. The proofs are obtained by giving theseidentites an algebraic explaination in terms of parabolic subgroups, coset representatives, and Poincar\'e series, and by a formal argument in terms of inclusion-exclusion-like matrices.
  • Special matchings and Coxeter groups (with F. Brenti and F. Caselli),
    Archiv der Mathematik, 89 (2007), no 4, 298-310

    In the recent paper [Adv. Applied Math., 38 (2007), 210--226] it is proved that the special matchings of permutations generate a Coxeter group. In this paper we generalize this result to a class of Coxeter groups which includes many Weyl and affine Weyl groups. Our proofs are simpler, and shorter, than those in {[}loc. cit.]
  • Algebraic and combinatorial properties of zircons,
    Journal of Algebraic Combinatorics, 26 (2007), no 3, 363-382

    In this paper we introduce and study a new class of posets, that we call zircons, which includes all Coxeter groups partially ordered by Bruhat order. We prove that many of the properties of Coxeter groups extend to zircons often with simpler proofs: in particular, zircons are Eulerian posets and the Kazhdan-Lusztig construction of the Kazhdan-Lusztig representations can be carried out in the context of zircons. Moreover, for any zircon $Z$, we construct and count all balanced and exact labelings (used in the construction of the Bernstein-Gelfand-Gelfand resolutions in the case that $Z$ is the Weyl group of a Kac-Moody algebra).
  • Special matchings and permutations in Bruhat orders (with F. Brenti and F. Caselli),
    Advances in Applied Mathematics, 38 (2007), 210-226

    In this paper we show that, for any permutation v, the special matchings of the lower Bruhat interval [e,v], considered as involutions on [e,v], generate a Coxeter system. This gives new necessary conditions on an abstract poset to be isomorphic to a lower Bruhat interval of the symmetric group, and also answers in the affirmative, in the symmetric group case, a problem posed in [F. Brenti, F. Caselli, M. Marietti, Advances in Math., 202 (2006), 555-601].
  • Diamonds and Hecke algebra representations (with F. Brenti and F. Caselli),
    International Mathematics Research Notices, vol. 2006, Article ID 29407, 34 pages, 2006

    In this work we show that Kazhdan and Lusztig's construction of Hecke algebra representations introduced in [D. Kazhdan, G. Lusztig, 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 Coxetergroups partially ordered by Bruhat order. We prove that one can define a family of polynomials indexed by pairs of elements in the diamond which reduce to the Kazhdan-Lusztig polynomials in the case that the diamond is a Coxeter group. We then show that every diamond contains in a natural way a Coxeter group and hence a Hecke algebra. Finally we show that this Coxeter group and the corresponding Hecke algebra act naturally on the diamond, and that the resulting representations include those constructed by Kazhdan and Lusztig, but contain several new ones.
  • Special matchings and Kazhdan-Lusztig polynomials (with F. Brenti and F. Caselli),
    Advances in Mathematics, 202 (2006), 555-601

    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 Cloux, Advances in Math., 180 (2003), 146-175] on the combinatorial invariance of Kazhdan-Lusztig polynomials.
  • Formulas for multi-parameter classes of Kazhdan-Lusztig polynomials in S(n) (with F. Caselli),
    Discrete Mathematics, 306 (2006), 711-725

    In this paper we give explicit formulas for several classes of Kazhdan-Lusztig polynomials of the symmetric group which are related to others already considered in the literature. In particular, we generalize two theorems of  Brenti and Simion  [F. Brenti, R. Simion, Explicit formulae for someKazhdan-Lusztig polynomials, J. Algebraic Combin. {11} (2000) 187--196], we prove an unpublished conjecture of Brenti, and we treat the case missing in Theorem 4.8 of [F. Caselli, {Proof of two conjectures of Brenti and Simion on Kazhdan-Lusztig polynomials}, J. Algebraic Combin. 18 (2003), 171--187].
  • Boolean elements in Kazhdan-Lusztig theory,
    Journal of  Algebra, 295 (2006), no 1, 1-26

    Kazhdan-Lusztig polynomials 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 Kazhdan-Lusztig polynomials indexed by Boolean elements in a class of Coxeter systems that we call linear. Boolean elements are elements smaller than a reflection that admits a reduced expression of the form $s_{1}\ldots s_{n-1}s_{n}s_{n-1}\ldots s_{1}$ ($s_{i}\in S$, $s_{i}\neq s_{j}$ if $i\neq j$). Then we provide several applications of this result concerning  the combinatorial invariance of the Kazhdan-Lusztig polynomials, the classification of the pairs $(u,v)$ with $u\prec v$, the Kazhdan-Lusztig elements and the intersection homology Poincar\'e polynomials of the Schubert varieties.
  • Closed product formulas for certain R-polynomials,
    European Journal of Combinatorics, 23 (2002), 57-62

    $R$-polynomials get their importance from the fact that they are used to define and compute the Kazhdan-Lusztig polynomials, which have applications in several fields. In this paper we give a closed product formula for certain $R$-polynomials valid for every Coxeter group. This result implies a conjecture due to F. Brenti about the symmetric groups.

     

    Proceedings and chapters in books

    • Root polytopes of crystallographic root systems,
      in Combinatorial Methods in Topology and Algebra,
      B. Benedetti,E. Delucchi, L. Moci Editors, 2015, Springer
    • KazhdanLusztig polynomials, tight quotients and Dyck superpartition,
      Oberwolfach Report No. 15/2010, 861-863

    Other papers and publications

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