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Logarithmic Concavity of Schur and Related Polynomials

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LOGARITHMIC CONCAVITY OF SCHUR AND RELATED POLYNOMIALS JUNE HUH, JACOB P. MATHERNE, KAROLA MESZ´ AROS,´ AND AVERY ST. DIZIER ABSTRACT. We show that normalized Schur polynomials are strongly log-concave. As a conse- quence, we obtain Okounkov’s log-concavity conjecture for Littlewood–Richardson coefficients in the special case of Kostka numbers. 1. INTRODUCTION Schur polynomials are the characters of finite-dimensional irreducible polynomial represen- tations of the general linear group GLmpCq. Combinatorially, the Schur polynomial of a partition λ in m variables is the generating function µpTq µpTq µ1pTq µmpTq sλpx1; : : : ; xmq “ x ; x “ x1 ¨ ¨ ¨ xm ; T ¸ where the sum is over all Young tableaux T of shape λ with entries from rms, and µipTq “ the number of i’s among the entries of T; for i “ 1; : : : ; m. Collecting Young tableaux of the same weight together, we get µ sλpx1; : : : ; xmq “ Kλµ x ; µ ¸ where Kλµ is the Kostka number counting Young tableaux of given shape λ and weight µ [Kos82]. Correspondingly, the Schur module Vpλq, an irreducible representation of the general linear group with highest weight λ, has the weight space decomposition Vpλq “ Vpλqµ with dim Vpλqµ “ Kλµ: µ arXiv:1906.09633v3 [math.CO] 25 Sep 2019 à Schur polynomials were first studied by Cauchy [Cau15], who defined them as ratios of alter- nants. The connection to the representation theory of GLmpCq was found by Schur [Sch01]. For a gentle introduction to these remarkable polynomials, and for all undefined terms, we refer to [Ful97]. June Huh received support from NSF Grant DMS-1638352 and the Ellentuck Fund. Jacob Matherne received support from NSF Grant DMS-1638352 and the Association of Members of the Institute for Advanced Study. Karola Mesz´ aros´ received support from NSF Grant DMS-1501059, CAREER NSF Grant DMS-1847284 and a von Neumann Fellowship funded by the Friends of the Institute for Advanced Study. 1 2 JUNE HUH, JACOB P. MATHERNE, KAROLA MESZ´ AROS,´ AND AVERY ST. DIZIER We prove several log-concavity properties of Schur polynomials. An operator that turns generating functions into exponential generating functions will play an important role. This linear operator, denoted N, is defined by the condition µ µ1 µm µ x x1 xm m Npx q “ “ ¨ ¨ ¨ for all µ P N : µ! µ1! µm! Recall that a partition is a weakly decreasing sequence of nonnegative integers. Theorem 1 (Continuous). For any partition λ, the normalized Schur polynomial xµ Nps px ; : : : ; x qq “ K λ 1 m λµ µ! µ ¸ m is either identically zero or its logarithm is concave on the positive orthant R¡0. m m Let ei be the i-th standard unit vector in N . For µ P Z and distinct i; j P rms, we set µpi; jq “ µ ` ei ´ ej: We show that the sequence of weight multiplicities of Vpλq we encounter is always log-concave if we walk in the weight diagram along any root direction ei ´ ej. Theorem 2 (Discrete). For any partition λ and any µ P Nm, we have 2 Kλµ ¥ Kλµpi;jqKλµpj;iq for any i; j P rms. ν For partitions ν; κ, λ, the Littlewood–Richardson coefficient cκλ is given by the decomposition cν Vpκq b Vpλq » Vpνq` κλ : ν à ν When the skew shape ν{κ has at most one box in each column, cκλ is the Kostka number Kλµ, where µ “ ν ´ κ.1 Conversely, for any partition λ and any µ, we have ν Kλµ “ cκλ; n n where ν and κ are the partitions given by νi “ j“i µj and κi “ j“i`1 µj. Thus Theorem2 verifies a special case of Okounkov’s conjecture that the discrete function ř ř ν pν; κ, λq ÞÝÑ log cκλ 1The equality between the Littlewood–Richardson coefficient and the Kostka number follows from Pieri’s formula hµ1 px1; : : : ; xmq ¨ ¨ ¨ hµm px1; : : : ; xmq “ Kλµsλpx1; : : : ; xmq; λ ¸ where hµi is the µi-th complete symmetric function [Ful97, Section 6.1]. When ν{κ has at most one box in each column, the left-hand side is the skew Schur function sν{κ, given by the Littlewood–Richardson rule ν sν{κpx1; : : : ; xmq “ cκλsλpx1; : : : ; xmq: λ ¸ LOGARITHMIC CONCAVITY OF SCHUR AND RELATED POLYNOMIALS 3 is concave [Oko03, Conjecture 1].2 We point out that, for any fixed λ, the log-concavity of Kλµ along any direction is known to hold asymptotically. By [Hec82], the Duistermaat–Heckman measure obtained from the orbit of λ under SUm is a translate of the weak limit Kkλµδ 1 µ lim µ k ; kÑ8 K ř µ kλµ 1 where δ 1 is the point mass at µ. It follows from [Gra96] that, in this case, the density function k µ k ř of the Duistermaat–Heckman measure is log-concave.3 We refer to [BGR04, Section 3] for an exposition. In [BH19], the authors introduce Lorentzian polynomials as a generalization of volume poly- nomials in algebraic geometry and stable polynomials in optimization theory. See Section2 for a brief introduction. We show that normalized Schur polynomials are Lorentzian in the sense of [BH19], and deduce Theorems1 and2 from the Lorentzian property. Theorem 3. The normalized Schur polynomial Npsλpx1; : : : ; xmqq is Lorentzian for any λ. Using general properties of Lorentzian polynomials [BH19, Section 6], Theorem3 can be strengthened as follows. 1 ` Corollary 4. For any sequence of partitions λ ; : : : ; λ and any positive integers m1; : : : ; m`, ` k (1) the normalized product of Schur polynomials Np k“1 sλ px1; : : : ; xmk qq is Lorentzian, and ` k (2) the product of normalized Schur polynomials k±“1 Npsλ px1; : : : ; xmk qq is Lorentzian. ± We prove Theorem3 in Section2 in a more general context of Schubert polynomials, but the main idea is simple enough to be outlined here. The volume polynomial of an irreducible complex 4 projective variety Y , with respect to a sequence of nef divisor classes H “ pH1;:::; Hmq, is the homogeneous polynomial 1 vol px ; : : : ; x q “ px H ` ¨ ¨ ¨ ` x H qdim Y ; Y;H 1 m dim Y ! 1 1 m m »Y where the intersection product of Y is used to expand the integrand. Volume polynomials are prototypical examples of Lorentzian polynomials [BH19, Section 10]. To show that the normal- ized Schur polynomial of λ is a volume polynomial, we suppose that the partition λ has m parts, 2 The conjecture holds in the “classical limit” [Oko03, Section 3], but the general case is refuted in [CDW07]: p4n;3n;2n;1nq n ` 2 p8n;6n;4n;2nq n ` 5 c “ and c “ for all n: p3n;2n;1nqp2n;1n;1nq 2 p6n;4n;2nqp4n;2n;2nq 5 ´ ¯ ´ ¯ The same example shows that the log-concavity conjecture for parabolic Kostka numbers [Kir04, Conjecture 6.17] also fails. 3Let pM;!q be a symplectic manifold of dimension 2n with an action of a torus T and a moment map M Ñ t˚. The Duistermaat–Heckman measure is the push-forward of the Liouville measure !n via the moment map. In this generality, Karshon shows that the density function need not be log-concave [Kar96].³ 4A Cartier divisor on a complete variety Y is nef if it intersects every curve in Y nonnegatively. We refer to [Laz04] for a comprehensive introduction. 4 JUNE HUH, JACOB P. MATHERNE, KAROLA MESZ´ AROS,´ AND AVERY ST. DIZIER and choose a large integer ` to get a complementary pair of partitions λ “ pλ1; λ2; : : : ; λmq and κ “ p`; `; : : : ; `q ´ pλm; λm´1; : : : ; λ1q: The Schur polynomials of the partitions λ and κ are related by the identity5 ` ` ´1 ´1 sκpx1; : : : ; xmq “ x1 ¨ ¨ ¨ xmsλpx1 ; : : : ; xm q: Let X be the product of projective spaces pP`qm, and let Y be a subvariety of X whose funda- mental class satisfies ˚ rY s “ sκpH1;:::; Hmq X rXs; Hi “ c1pπi Op1qq; where πi is the i-th projection. The volume polynomial of Y with respect to H is 1 vol px ; : : : ; x q “ px H ` ¨ ¨ ¨ ` x H qdim Y Y;H 1 m dim Y ! 1 1 n m »Y 1 “ s pH ;:::; H qpx H ` ¨ ¨ ¨ ` x H qdim Y “ Nps px ; : : : ; x qq: dim Y ! κ 1 m 1 1 m m λ 1 m »X m ˚ Such Y can be constructed from a sequence of generic global sections i“1 πi Op1q as a degen- eracy locus [Ful98, Example 14.3.2], completing the argument. À In Section2, we introduce Lorentzian polynomials and prove the main results. In Section3, we present evidence for the ubiquity of Lorentzian polynomials through a series of results and conjectures. Acknowledgments. We are grateful to Dave Anderson, Alex Fink, Allen Knutson, Thomas Lam, Ricky Liu, Alex Postnikov, Pavlo Pylyavskyy, Vic Reiner, Mark Shimozono, and Alex Yong for fruitful discussions. We thank the Institute for Advanced Study for providing an excellent environment for our collaboration. 2. NORMALIZED SCHUR POLYNOMIALS ARE LORENTZIAN A subset J Ď Zn is M-convex6 if, for any index i P rns and any α P J and β P J whose i-th coordinates satisfy αi ¡ βi, there is an index j P rns satisfying αj ă βj and α ´ ei ` ej P J and β ´ ej ` ei P J: The notion of M-convexity forms the foundation of discrete convex analysis [Mur03]. The con- vex hull of an M-convex set is a generalized permutohedron in the sense of [Pos09], and con- versely, the set of integral points in an integral generalized permutohedron is an M-convex set [Mur03, Theorem 1.9]. Lorentzian polynomials connect discrete convex analysis with many log-concavity phenom- ena in combinatorics. See [AOGV18, ALOGV18a, ALOGV18b, BES19, BH18, BH19, EH19] for 5 The dual of the Schur module Vpλq has highest weight p´λm;:::; ´λ1q, see [FH91, Exercise 15.50].
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  • The Permutahedral Variety, Mixed Eulerian Numbers, and Principal Specializations of Schubert Polynomials

    The Permutahedral Variety, Mixed Eulerian Numbers, and Principal Specializations of Schubert Polynomials

    THE PERMUTAHEDRAL VARIETY, MIXED EULERIAN NUMBERS, AND PRINCIPAL SPECIALIZATIONS OF SCHUBERT POLYNOMIALS PHILIPPE NADEAU AND VASU TEWARI Abstract. We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants aw are expressed as a sum of normalized mixed Eulerian numbers indexed naturally by reduced words of w. The description implies that the aw are positive for all permutations w 2 Sn of length n − 1, thereby answering a question of Harada, Horiguchi, Masuda and Park. We use the same expression to establish the invariance of aw under taking inverses and conjugation by the longest word, and subsequently establish an intriguing cyclic sum rule for the numbers. We then move toward a deeper combinatorial understanding for the aw by exploiting in addition the relation to Postnikov's divided symmetrization. Finally, we are able to give a combinatorial interpretation for aw when w is vexillary, in terms of certain tableau descents. It is based in part on a relation between the numbers aw and principal specializations of Schubert polynomials. Along the way, we prove results and raise questions of independent interest about the combina- torics of permutations, Schubert polynomials and related objects. We also sketch how to extend our approach to other Lie types, highlighting in particular an identity of Klyachko. Contents 1. Introduction and statement of results 2 2. Preliminaries 6 3. Formulas for aw 11 4. Mixed Eulerian numbers 13 5. Properties of the numbers aw 16 6. Combinatorial interpretation of aw in special cases 20 7. The case of vexillary permutations 25 8.