We show that five nonnegativity properties of polynomials coincide when restricted to polynomials of the form x1, pi(1) ... xn,pi(n) - x1, sigma(1) ... xn, sigma(n), where $\pi and sigma are permutations in Sn. In particular, we show that each of these properties may be used to characterize the Bruhat order on Sn

Drake, Brian

Gerrish, Sean

Skandera, Mark

English

Dartmouth Digital Commons (Dartmouth College)

Dartmouth CollegeDartmouth Digital CommonsOpen Dartmouth: Faculty Open Access Articles6-3-2005Monomial Nonnegativity and the Bruhat OrderBrian DrakeBrandeis UniversitySean GerrishUniversity of Michigan-Ann ArborMark SkanderaDartmouth CollegeFollow this and additional works at: https://digitalcommons.dartmouth.edu/facoaPart of the Discrete Mathematics and Combinatorics CommonsThis Article is brought to you for free and open access by Dartmouth Digital Commons. It has been accepted for inclusion in Open Dartmouth: FacultyOpen Access Articles by an authorized administrator of Dartmouth Digital Commons. For more information, please contactdartmouthdigitalcommons@groups.dartmouth.edu.Recommended CitationDrake, Brian; Gerrish, Sean; and Skandera, Mark, "Monomial Nonnegativity and the Bruhat Order" (2005). Open Dartmouth: FacultyOpen Access Articles. 2326.https://digitalcommons.dartmouth.edu/facoa/2326Monomial nonnegativity and the Bruhat orderBrian Drake, Sean Gerrish, Mark SkanderaDept. of Mathematics, Brandeis UniversityMS 050, P.O. Box 9110, Waltham, MA 02454bdrake@math.brandeis.eduDept. of Mathematics, University of Michigan2074 East Hall, Ann Arbor, MI 48109-1109sgerrish@umich.eduDept. of Mathematics, Dartmouth College6188 Bradley Hall, Hanover, NH 03755-3551mark.skandera@dartmouth.eduSubmitted: Mar 11, 2005; Accepted: May 6, 2005; Published: Jun 3, 2005MR Subject Classications: 15A15, 05E05AbstractWe show that ve nonnegativity properties of polynomials coincide when re-stricted to polynomials of the form x1;(1)    xn;(n) − x1;(1)    xn;(n), where and  are permutations in Sn. In particular, we show that each of these propertiesmay be used to characterize the Bruhat order on Sn.1 IntroductionLet x = (xij) be a generic square matrix and dene I;I′(x) to be the (I; I0) minor of x,i.e., the determinant of the submatrix of x corresponding to rows I and columns I 0. Areal matrix is called totally nonnegative (TNN) if each of its minors is nonnegative. (Seee.g. [9].) A polynomial p(x11; : : : ; xnn) in n2 variables is called totally nonnegative if itsatisesp(A) =defp(a1;1; : : : ; an;n)  0 (1)for each n  n totally nonnegative matrix A = (ai;j). Some recent interest in totalnonnegativity concerns a set of polynomials known in quantum Lie theory as the dualcanonical basis of O(SL(n;C)). (See e.g. [25].) In particular, Lusztig [17] has proved thatthese polynomials are TNN.A polynomial p(x) which is equal to a subtraction-free rational expression in matrixminors must be TNN. (By a result of Whitney [24], we need not be concerned that thedenominator vanishes for some TNN matrices.) We shall say that such a polynomial p(x)has the subtraction-free rational function (SFR) property. If this subtraction-free rationalthe electronic journal of combinatorics 11(2) (2005), #R18 1expression may be chosen so that the denominator is a monomial in matrix minors, weshall say that p(x) has the subtraction-free Laurent (SFL) property. One example of apolynomial having the SFL property isx1;2x2;1x3;3 − x1;2x2;3x3;1 − x1;3x2;1x3;2 + x1;3x2;2x3;1=13;23(x)23;13(x) + 1;3(x)3;1(x)23;23(x)3;3(x):Analogous classes of polynomials may be dened in terms of symmetric functions.(See [21, Ch. 7] for basic denitions concerning symmetric functions.) In particular, anynite submatrix of the innite matrix H = (hj−i)i;j0, where hk is the kth complete ho-mogeneous symmetric function and hk = 0 for k < 0, is called a Jacobi-Trudi matrix.We dene a polynomial p(x1;1; : : : ; xn;n) to be monomial nonnegative (MNN) if for eachJacobi-Trudi matrix A = (ai;j) the symmetric function p(A) is equal to a nonnegativelinear combination of monomial symmetric functions. Dening Schur nonnegative (SNN)polynomials analogously, we have that every SNN polynomial is MNN. Some recent in-terest in SNN polynomials is motivated by problems in algebraic geometry [8, Conj. 2.8,Conj. 5.1], [1].2 Main resultThe ve nonnegativity properties dened in Section 1 have been applied most often toimmanants, polynomials which belong to spanCfx1;(1)   xn;(n) j  2 Sng. (See [11], [12],[13], [20], [19], [22], [23]. The results of [7] may also be stated in these terms.) Curiously,the TNN, MNN, and SNN properties coincide when applied to immanants in the maintheorems of the above papers. It is also curious that none of these immanants is knownnot to have the SFL property. It would be interesting to identify immanants which havesome of these nonnegativity properties and fail to have others. Nevertheless, our mainresult shows that the ve properties coincide when applied to immanants of the formx1;(1)   xn;(n) − x1;(1)   xn;(n):We shall use the following well-known characterizations of the Bruhat order on Sn.The Bruhat order on Sn is often dened by comparing two permutations  = (1)   (n)and  = (1)   (n) according to the following criterion:    if  is obtainable from by a sequence of transpositions (i; j) where i < j and i appears to the left of j in . (Seee.g. [14, p. 119].) A second well-known criterion compares permutations in terms of theirdening matrices. Let M() be the matrix whose (i; j) entry is 1 if j = (i) and zerootherwise. Dening [i] = f1; : : : ; ig, and denoting the submatrix of M() correspondingto rows I and columns J by M()I;J , we have the following.Theorem 1 Let  and  be two permutations in Sn. Then  is less than or equal to  inthe Bruhat order if and only if for all 1  i; j  n− 1, the number of ones in M()[i];[j]is greater than or equal to the number of ones in M()[i];[j].the electronic journal of combinatorics 11(2) (2005), #R18 2(See [2], [3], [4], [6], [10, pp. 173-177], [16], [15], [18]. for more characterizations.)Our result, combined with those of our previous paper [5], is the following list ofnonnegativity criteria with which one may dene the Bruhat order.Theorem 2 Let  and  be permutations in Sn. The following conditions on  and are equivalent.1.    in the Bruhat order.2. x1;(1)   xn;(n) − x1;(1)   xn;(n) is totally nonnegative.3. x1;(1)   xn;(n) − x1;(1)   xn;(n) is Schur nonnegative.4. x1;(1)   xn;(n) − x1;(1)   xn;(n) is monomial nonnegative.5. x1;(1)   xn;(n)−x1;(1)   xn;(n) has the subtraction-free rational function property.6. x1;(1)   xn;(n) − x1;(1)   xn;(n) has the subtraction-free Laurent property.Proof: The implications (3 ) 4) and (6 ) 5 ) 2) are immediate. The implication(2 ) 1) was estblished in [5, Thm. 2], and the implication (1 ) 6) follows trivially fromthat proof. The implication (1 ) 3) was established in [5, Thm. 3]. It will suce thereforeto prove the implication (4 ) 1).Suppose that  is not less than or equal to  in the Bruhat order. By Theorem 1 wemay choose indices 1  k; `  n−1 such that M()[k];[`] contains q+1 ones and M()[k];[`]contains q ones. Keeping n xed, let b be a large nonnegative integer which satises2bb> (2b + 2n)2n2;(which is possible because(2bbgrows exponentially) and consider the n n Jacobi-TrudimatrixB =266666664hb+k−1    hb+k+`−2 h2b+k−1    h2b+n+k−`−2............hb    hb+`−1 h2b    h2b+n−1−`hn−k−1    hn−k+`−2 hb+n−k−1    hb+2n−k−`−1............h0    h`−1 hb    hb+n−`−1377777775;dened by the skew shape (2b + k − `− 1)k(b + n− `− 1)n−k=(b− `)`. Lets = k(2b + k − `− 1) + (n− k)(b + n− `− 1)− `(b− `)be the number of boxes in this skew shape.The polynomial x1;(1)   xn;(n) − x1;(1)   xn;(n) applied to B may be expressed ash − h for some appropriate partitions ;  of s, which depend on ; , respectively. Weclaim that the coecient of m1s in the monomial expansion of h − h is negative.the electronic journal of combinatorics 11(2) (2005), #R18 3Note that the ratio of the coecients of m1s in the monomial expansions of h and his (s1;:::;n(s1;:::;n = 1!   n!1!   n! :By the locations of ones in the matrices M() and M(), this ratio is less than or equalto(2b + 2n)!k−q−1(2b)!k−q(b + 2n)!n−k−`+2q+2b!n−k−`+2q(2n)!`−q−10!`−q;which in turn is less than or equal to(2b + 2n)2n(k−q−1)(2b)!(b + 2n)!2(2b + 2n)2n(n−k+q−1) =(b + 2n)!2(2b)!(2b + 2n)2n(n−2) (2b + 2n)2n2(2bb ;which is less than 1 by our choice of b. It follows that the coecient of m1s in the monomialexpansion of h − h is negative and the polynomial x1;(1)   xn;(n) − x1;(1)   xn;(n) isnot MNN. AcknowledgementsThe authors are grateful to Arun Ram and an anonymous referee for helpful conversations.References[1] F. Bergeron, R. Biagioli, and M. Rosas. Inequalities between Littlewood-Richardson coecients (2004). Preprint math.CO/0403541 on ArXiv.[2] A. Bjo¨rner. Orderings of Coxeter groups. In Combinatorics and Algebra(C. Greene, ed.), vol. 34 of Contemp. Math.. American Mathematical Society, Prov-idence, RI (1984), pp. 175{195.[3] A. Bjo¨rner and F. Brenti. An improved tableau criterion for Bruhat order.Electron. J. Combin., 3, 1 (1996). Research paper 22, 5 pp. (electronic).[4] V. Deodhar. Some characterizations of Bruhat ordering on a Coxeter group anddetermination of the relative Mo¨bius function. Inventiones Math., 39 (1977) pp.187{198.[5] B. Drake, S. Gerrish, and M. Skandera. Two new criteria for comparison inthe Bruhat order. Electron. J. Combin., 11, 1 (2004). Note 6, 4 pp. (electronic).[6] C. Ehresmann. Sur la topologie de certains espaces homogenes. Ann. Math., 35(1934) pp. 187{198.[7] S. M. Fallat, M. I. Gekhtman, and C. R. Johnson. Multiplicative principal-minor inequalities for totally nonnegative matrices. Adv. Appl. Math., 30, 3 (2003)pp. 442{470.the electronic journal of combinatorics 11(2) (2005), #R18 4[8] S. Fomin, W. Fulton, C. K. Li, and Y. Poon. Eigenvalues, singular values,and Littlewood-Richardson coecients. Amer. J. Math., 127, 1 (2005) pp. 101{127.[9] S. Fomin and A. Zelevinsky. Total positivity: Tests and parametrizations. Math.Intelligencer , 22, 1 (2000) pp. 23{33.[10] W. Fulton. Young Tableaux; With Applications to Representation Theory and Ge-ometry , vol. 35 of London Mathematical Society Student Texts. Cambridge UniversityPress, New York (1997).[11] I. P. Goulden and D. M. Jackson. Immanants of combinatorial matrices. J.Algebra, 148 (1992) pp. 305{324.[12] C. Greene. Proof of a conjecture on immanants of the Jacobi-Trudi matrix. LinearAlgebra Appl., 171 (1992) pp. 65{79.[13] M. Haiman. Hecke algebra characters and immanant conjectures. J. Amer. Math.Soc., 6, 3 (1993) pp. 569{595.[14] J. E. Humphreys. Reflection groups and Coxeter groups. Cambridge UniversityPress (1990).[15] A. Lascoux. Potentiel Yin sur le groupe symetrique. Sem. Lothar. Combin., 38(1996). Art. B38a, 12 pp. (electronic).[16] A. Lascoux and M. P. Schu¨tzenberger. Treillis et bases des groupes de Cox-eter. Electron. J. Combin., 3, 2 (1996). Research paper 27, 35 pp. (electronic).[17] G. Lusztig. Total positivity in reductive groups. In Lie Theory and Geometry: inHonor of Bertram Kostant , vol. 123 of Progress in Mathematics . Birkha¨user, Boston(1994), pp. 531{568.[18] R. A. Proctor. Classical Bruhat orders and lexicographic shellability. J. Algebra,77 (1982) pp. 104{126.[19] B. Rhoades and M. Skandera. Kazhdan-Lusztig immanants and products ofmatrix minors (2005). In progress.[20] B. Rhoades and M. Skandera. Temperley-Lieb immanants. Ann. Comb. (2005).To appear.[21] R. Stanley. Enumerative Combinatorics , vol. 2. Cambridge University Press,Cambridge (1999).[22] R. Stanley and J. R. Stembridge. On immanants of Jacobi-Trudi matrices andpermutations with restricted positions. J. Combin. Theory Ser. A, 62 (1993) pp.261{279.[23] J. Stembridge. Some conjectures for immanants. Can. J. Math., 44, 5 (1992) pp.1079{1099.[24] A. Whitney. A reduction theorem for totally positive matrices. J. Anal. Math., 2(1952) pp. 88{92.[25] A. Zelevinsky. From Littlewood-Richardson coecients to cluster algebras in threelectures. In Symmetric Functions 2001: Surveys of Developments and Perspectives(S. Fomin, ed.), vol. 74 of NATO Science Series II: Mathematics, Physics, andChemistry . Kluwer, Dordrecht (2002), pp. 253{273.the electronic journal of combinatorics 11(2) (2005), #R18 5

Monomial Nonnegativity and the Bruhat Order

https://digitalcommons.dartmouth.edu/cgi/viewcontent.cgi?article=3330&amp;context=facoa

Monomial Nonnegativity and the Bruhat Order

Abstract

Similar works

Full text

Available Versions

Dartmouth Digital Commons (Dartmouth College)