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Divided Symmetrization and Quasisymmetric Functions (Extended Abstract) Philippe Nadeau, Vasu Tewari

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Divided Symmetrization and Quasisymmetric Functions (Extended Abstract) Philippe Nadeau, Vasu Tewari Divided symmetrization and quasisymmetric functions (extended abstract) Philippe Nadeau, Vasu Tewari To cite this version: Philippe Nadeau, Vasu Tewari. Divided symmetrization and quasisymmetric functions (extended abstract). Seminaire Lotharingien de Combinatoire, Université Louis Pasteur, 2020. hal-03099482 HAL Id: hal-03099482 https://hal.archives-ouvertes.fr/hal-03099482 Submitted on 13 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Séminaire Lotharingien de Combinatoire XX (2020) Proceedings of the 32nd Conference on Formal Power Article #YY, 12 pp. Series and Algebraic Combinatorics (Ramat Gan) Divided symmetrization and quasisymmetric functions Philippe Nadeau∗1, and Vasu Tewariy2 1 Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR 5208, Institut Camille Jordan, France 2 Department of Mathematics, University of Pennsylvania, USA Abstract. We study various aspects of the divided symmetrization operator, which was introduced by Postnikov in the context of volume polynomials of permutahe- dra. Divided symmetrization is a linear form which acts on the space of polynomials in n indeterminates of degree n − 1. Our main results are related to quasisymmet- ric polynomials. We show that divided symmetrization applied to a quasisymmetric polynomial in m ≤ n indeterminates has a natural interpretation. We further show that divided symmetrization of any polynomial can be naturally computed with respect to a direct sum decomposition due to Aval-Bergeron-Bergeron, involving the ideal gen- erated by positive degree quasisymmetric polynomials in n indeterminates. Our main motivation for studying divided symmetrization comes from studying the cohomology class of the Peterson variety. Keywords: divided symmetrization, quasisymmetric functions 1 Introduction Toward computing volume polynomials of permutahedra, Postnikov [10] introduced an operator called divided symmetrization (abbreviated to DS in the sequel on occa- sion). It takes a polynomial f (x1,..., xn) as input and outputs a symmetric polynomial ( ) f x1,..., xn n defined by f (x ,..., x ) ( ) = · 1 n f x1,..., xn n : ∑ w , ∏ (x − x + ) w2Sn 1≤i≤n−1 i i 1 where Sn denotes the symmetric group on n letters, naturally acting by permuting vari- ables. It can be checked that divided symmetrization acting on homogeneous polyno- mials of degree strictly less than n − 1 results in 0. Thus the degree n − 1 case is the first non-trivial case, and in this context divided symmetrizations results in scalars. ∗[email protected]. [email protected]. Vasu Tewari acknowledges financial support of the AMS-Simons travel grant. 2 Philippe Nadeau, and Vasu Tewari n A pertinent and motivating instance is the following. Given a = (a1,..., an) 2 R , the permutahedron Pa is the convex hull of all points of the form (aw(1),..., aw(n)) where w ranges over all permutations in Sn. Postnikov [10, Section 3] shows that the volume of P 1 ( + ··· + )n−1 a is given by (n−1)! a1x1 anxn n. While a great deal of research has been conducted into various aspects of permutahe- dra, especially in regard to volumes and lattice point enumeration, divided symmetriza- tion has received limited attention; see [1,9]. Our own motivation for studying divided symmetrization stems from a problem in Schubert calculus, which we sketch now. The flag variety Fl(n) is a complex projective variety structure on the set of complete flags, which are sequences F0 = f0g ⊂ F1 ⊂ n F2 ⊂ · · · ⊂ Fn−1 ⊂ Fn = C of subspaces such that dim Fi = i. The Schubert varieties Xw ⊂ Fl(n), for w 2 Sn, give rise to the basis of Schubert classes sw in the integral ∗ cohomology H (Fl(n)). The Peterson variety Petn is the subvariety of Fl(n) of dimension n − 1 comprised of flags (Fi)i=0,...,n such that NFi ⊂ Fi+1 for i < n, where N is the n × n matrix with ones above the diagonal and zeros everywhere else. It is a special case of a regular nilpotent Hessenberg variety. Our initial problem was to compute the number aw of points in the intersection of Petn with a generic translate of a Schubert variety Xw, for w of length n − 1. Equivalently, ∗ aw is the coefficient of the class [Petn] 2 H (Fl(n)) on the class sw. We can then show S ( ) S that aw is given by w x1,..., xn n where w is the celebrated Schubert polynomial attached to w. The results presented in this extended abstract are thus primarily motivated by un- derstanding the divided symmetrization of Schubert polynomials. The reader is referred to [8] for the version with proofs. We aim to understand more about the structure of the divided symmetrization operator acting on polynomials of degree n − 1, since Post- nikov’s work and our own work coming from Schubert calculus both have this condition. Our investigations allows us to uncover a direct (and intriguing) connection between di- vided symmetrization and quasisymmetric polynomials. The ring of quasisymmetric functions in infinitely many variables x = fx1, x2,... g was introduced by Gessel [5] and has since acquired great importance in algebraic com- binatorics. A distinguished linear basis for this ring is given by the fundamental qua- sisymmetric functions Fa where a is a composition. Given a positive integer n, consider a quasisymmetric function f (x) of degree n − 1. We denote the quasisymmetric polyno- mial obtained by setting xi = 0 for all i > m by f (x1,..., xm) and refer to the evaluation m of f (x1,..., xm) at x1 = ··· = xm = 1 by f (1 ). Our first main result states the following: Theorem 1. For a quasisymmetric function f of degree n − 1, we have n m ∑m= f (x1,..., xm) t f (1j)tj = 0 n . ∑ ( − )n j≥0 1 t Divided symmetrization and quasisymmetric functions 3 A natural candidate for f comes from Stanley’s theory of P-partitions [11, 12]: To any naturally labeled poset P on n − 1 elements, one can associate a quasisymmetric function KP(x) with degree n − 1. Let L(P) denote the set of linear extensions of P. Note that elements in L(P) are permutations in Sn−1. Under this setup, we obtain the following corollary of Theorem1. Corollary 1. For m ≤ n, we have ( ) = jf 2 L( ) j − gj KP x1,..., xm n p P p has m 1 descents . We further establish connections between a quotient ring of polynomials investigated by [3,2]. Let Jn denote the ideal in Q[xn] := Q[x1,..., xn] generated by homogeneous quasisymmetric polynomials in x1,..., xn of positive degree. Let Rn be the degree n − 1 homogeneous component of Q[xn] , and let Kn := Rn \Jn. Aval-Bergeron-Bergeron [2] † provide a distinguished basis for a certain complementary space Kn of Kn in Rn, defined in Section5. This leads to our second main result. 2 = 2 = + Theorem 2. If f Kn, then f n 0. More generally, if f Rn is written f g h with † g 2 Kn and h 2 Kn according to (5.1), then = ( ) f n g 1, . , 1 . Outline of the article: Section2 sets up the necessary notations and definitions. In Section3 we discuss the case of monomials of degree n − 1 and define our distinguished class of Catalan compositions that plays a crucial role. In Section4 we focus on qua- sisymmetric polynomials, beginning with the basis of quasisymmetric monomials. Our central result stated above as Theorem1 is proved there. Section5 deepens the connec- tion with quasisymmetric polynomials by way of Theorem5. 2 Background Throughout, for a nonnegative integer n, we set [n] := fi j 1 ≤ i ≤ ng. In particular, [0] = Æ. We denote the set of variables fx1,..., xng by xn. Furthermore, set Q[xn] := Q[x1,..., xn]. We refer the reader to [11, 12] for any undefined terminology. 2.1 Compositions Given a nonnegative integer k and a positive integer n, a weak composition of k with n parts is a sequence (c1,..., cn) of nonnegative integers whose sum is k. We denote the 0 set of compositions n − 1, which play a special role in what follows, by Wn. The size of a weak composition c = (c1,..., cn) is the sum of its parts and is denoted by jcj.A 4 Philippe Nadeau, and Vasu Tewari strong composition is a weak composition all of whose parts are positive. Given a weak composition c, we denote the underlying strong composition obtained by omitting zero parts by c+. Henceforth, by the term composition, we always mean strong composition. If the size of a composition a is k, we denote this by a k. We denote the number of parts of a by `(a). Given a = (a1,..., a`(a)) k for k a positive integer, we associate a subset Set(a) = fa1, a1 + a2,..., a1 + ··· + a`(a)−1g ⊆ [k − 1]. This establishes a bijection between com- positions of k and subsets of [k − 1]. Given S ⊆ [k − 1], we define comp(S) to be the composition of k associated to S under the preceding correspondence. The inclu- sion order on subsets allows us to define the refinement order on compositions. More specifically, given a and b both compositions of k, we say that b refines a, denoted by a 4 b, if Set(a) ⊆ Set(b). For instance, we have a = (1, 3, 2, 2) 4 (1, 2, 1, 1, 1, 2) = b as Set(a) = f1, 4, 6g is a subset of Set(b) = f1, 3, 4, 5, 6g.
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