Printer Friendly

An algebraic proof of determinant formulas of Grothendieck polynomials.

1. Definition and Theorems. In [12] and [14], Lascoux and Schutzenberger introduced (double) Grothendieck polynomials indexed by permutations as representatives of K-theory classes of structure sheaves of Schubert varieties in a full flag variety. In [4] and [5], Fomin and Kirillov introduced [beta]-Grothendieck polynomials in the framework of Yang-Baxter equations together with their combinatorial formula and showed that they coincide with the ones defined by Lascoux and Schutzenberger with the specialization [beta] = - 1. Let x = ([x.sub.1], ..., [x.sub.d]), b = ([b.sub.1], [b.sub.2], ...) be sets of indetermiants. A Grassmannian permutation with descent at d corresponds to a partition [lambda] of length at most d, i.e. a sequence of non-negative integers [lambda] = ([[lambda].sub.1], ..., [[lambda].sub.d]) such that [[lambda].sub.i] [greater than or equal to] [[lambda].sub.i+1] for each i = 1, ..., d - 1. For such permutation, Buch [3] gave a combinatorial expression of the corresponding Grothendieck polynomial [G.sub.[lambda]](x) as a generating series of set-valued tableaux, a generalization of semi-standard Young tableaux by allowing a filling of a box in the Young diagram to be a set of integers. In [18], McNamara gave an expression of factorial (double [beta]-) Grothendieck polynomials [G.sub.[lambda]](x|b) also in terms of set-valued tableaux.

In this paper, we prove the following Jacobi Trudi type determinant formulas for [G.sub.[lambda]](x|b). For each non-negative integer k and an integer m, let [G.sup.(k).sub.m](x|b) be a function of x and b given by

[mathematical expression not reproducible],

where [beta] is a formal variable of degree -1 and 1/[1 + [beta][u.sup.-1]] is expanded as [[summation].sub.s[greater than or equal to]0][(-1).sup.s][[beta].sup.s][u.sup.-s]. We use the generalized binomial coefficients [mathematical expression not reproducible] given by [mathematical expression not reproducible] for n [member of] Z with the convention that [mathematical expression not reproducible] for all integers i < 0.

Theorem 1.1. For each partition [lambda] of length at most d, we have

[mathematical expression not reproducible].

Theorem 1.2. We have

[mathematical expression not reproducible].

In particular, we have

[mathematical expression not reproducible].

Theorems 1.1 and 1.2 were originally obtained in the context of degeneracy loci formulas for flag bundles by Hudson Matsumura in [7] and Hudson Ikeda-Matsumura-Naruse in [6] respectively. The proof in this paper is purely algebraic, generalizing Macdonald's argument in [16, (3.6)] for Jacobi-Trudi formula of Schur polynomials. It is based on the following "bi-alternant" formula of [G.sub.[lambda]](x|b) described by Ikeda-Naruse in [8]:

(1) [mathematical expression not reproducible].

Here we denote x [direct sum] y := x + y + [beta]xy and [[y|b].sup.k] := (y [direct sum] [b.sub.1]) ... (y [direct sum] [b.sub.k]) for any variable x, y. Note that the Grothendieck polynomial [G.sub.[lambda]](x) given in [3] coincides with [G.sub.[lambda]](x|b) by setting [beta] = - 1 and [b.sub.i] = 0

Determinant formulas different from the ones in Theorems 1.1 and 1.2 have been also obtained by Lenart in [15] (cf. [2], [13]), by Kirillov in [10] and [11], and by Yeliussizov [22]. Each entry of these previously known determinant formulas is given as a finite linear combination of elementary/ complete symmetric polynomials, while in our formula it is given as a possibly infinite linear combination of Grothendieck polynomials associated to one row partitions. A combinatorial proof of Theorem 1.2 has been also obtained in [17] for the non-factorial case, as well as an analogous determinant formula for skew flagged Grothendieck polynomials, special cases of which arise as the Grothendieck polynomials associated to 321-avoiding permutations [1] and vexillary permutations.

It is also worth mentioning that in [3] Buch obtained the Littlewood-Richardson rule for the structure constants of Grothendieck polynomials [G.sub.[lambda]](x), and hence the Schubert structure constants of the K-theory of Grassmannians (see also the paper [9] by Ikeda-Shimazaki for another proof) For the equivariant K-theory of Grassmannians (or equivalently for [G.sub.[lambda]](x|b)), the structure constants were determined by Pechenik and Yong in [20] by introducing a new combinatorial object called genomic tableaux. Motegi Sakai [19] identified Grothendieck polynomials with the wave functions arising in the five vertex models and obtained a variant of the Cauchy identity. Using this framework of integrable systems, Wheeler-Zinn-Justin [21] recently obtained another equivariant Littlewood-Richardson rule for factorial Grothendieck polynomials.

2. Proof of Theorem 1.1. By (1), it suffices to show the identity

[mathematical expression not reproducible],

for each ([a.sub.1], ..., [a.sub.d]) [member of] [Z.sup.d] such that [a.sub.i] + d - i [greater than or equal to] 0. For each j = 1, ..., d, we let

[mathematical expression not reproducible].

We denote [bar.y] := -y/[1 + [beta]y]. Since 1 + (u + [beta])y = [1 - [bar.y]u]/[1 + [beta][bar.y]], we have

[mathematical expression not reproducible].

Consider the identity

[mathematical expression not reproducible].

By comparing the coefficient of [u.sup.m], m [greater than or equal to] k in (2) we obtain

[mathematical expression not reproducible].

Since [y - [bar.b]]/[1 + [beta][bar.b]] = y [direct sum] b, we have

(2) [mathematical expression not reproducible].

Consider the matrices

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

By using (2), we find that the (i, j)-entry of HM is

[mathematical expression not reproducible].

By taking the determinant of HM, the factor [mathematical expression not reproducible] which turns to be 1 comes out, and therefore we obtain

[mathematical expression not reproducible].

By dividing by det M, we obtain the desired identity since det M = [[PI].sub.1[less than or equal to]i<j[less than or equal to]d]([x.sub.i] - [x.sub.j]) (see [16, p. 42]).

3. Proof of Theorem 1.2. By (1), it suffices to show the identity

[mathematical expression not reproducible]

for each ([a.sub.1], ..., [a.sub.d]) [member of] [Z.sup.d] such that [a.sub.i] + d - i [greater than or equal to] 0. For each j = 1, ..., d, let

[mathematical expression not reproducible].

Since 1 + (u + [beta])y = [1 - [bar.y]u]/[1 + [beta][bar.y]], we have the identity

(3) [mathematical expression not reproducible].

By comparing the coefficient of [u.sup.m], m [greater than or equal to] k in (3) we obtain

(4) [mathematical expression not reproducible]

where the last equality follows from the identity [x - [bar.y]]/[1 + [beta][bar.y]] = x [direct sum] y for any variable x, y.

Consider the matrices

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

We write the (i, j)-entry of the product H'[bar.M] as

[mathematical expression not reproducible].

By writing [mathematical expression not reproducible] using a well-known identity of binomial coefficients and then applying (4), we obtain

[mathematical expression not reproducible].

By taking the determinant of H'[bar.M], we have

[mathematical expression not reproducible].

Since we have (see [16, p. 42])

[mathematical expression not reproducible],

we obtain the desired identity.

doi: 10.3792/pjaa.93.82

Acknowledgements. The author would like to thank Prof. Takeshi Ikeda for useful discussions, and the referee for valuable comments. The author is supported by Grant-in-Aid for Young Scientists (B) 16K17584.

References

[1] D. Anderson, L. Chen and N. Tarasca, K-classes of Brill-Noether loci and a determinantal formula, arXiv:1705.02992.

[2] A. S. Buch, Grothendieck classes of quiver varieties, Duke Math. J. 115 (2002), no. 1, 75-103.

[3] A. S. Buch, A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), no. 1, 37-78.

[4] S. Fomin and A. N. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, in Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993), Discrete Math. 153 (1996), no. 1-3, 123-143.

[5] S. Fomin and A. N. Kirillov, Grothendieck polynomials and the Yang-Baxter equation, in Formal power series and algebraic combinatorics/Series formelles et combinatoire algebrique (University of Rutgers, Piscataway, 1994), 183-189, DIMACS, Piscataway, NJ, s.d.

[6] T. Hudson, T. Ikeda, T. Matsumura and H. Naruse, Degeneracy loci classes in K-theory --Determinantal and Pfaffian formula--, arXiv:1504.02828.

[7] T. Hudson and T. Matsumura, Segre classes and Kempf-Laksov formula in algebraic cobordism, arXiv:1602.05704.

[8] T. Ikeda and H. Naruse, K-theoretic analogues of factorial Schur P- and Q-functions, Adv. Math. 243 (2013), 22-66.

[9] T. Ikeda and T. Shimazaki, A proof of K-theoretic Littlewood-Richardson rules by Bender-Knuthtype involutions, Math. Res. Lett. 21 (2014), no. 2, 333-339.

[10] A. N. Kirillov, On some algebraic and combinatorial properties of Dunkl elements, Internat. J. Modern Phys. B 26 (2012), no. 27 28, 1243012, 28 pp.

[11] A. N. Kirillov, On some quadratic algebras I 1/2: combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 002, 172 pp.

[12] A. Lascoux, Anneau de Grothendieck de la variete de drapeaux, in The Grothendieck Festschrift, Vol. III, Progr. Math., 88, Birkhauser Boston, Boston, MA, 1990, pp. 1-34.

[13] A. Lascoux and H. Naruse, Finite sum Cauchy identity for dual Grothendieck polynomials, Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 7, 87-91.

[14] A. Lascoux and M.-P. Schutzenberger, Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variete de drapeaux, C. R. Acad. Sci. Paris Ser. I Math. 295 (1982), no. 11, 629-633.

[15] C. Lenart, Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb. 4 (2000), no. 1, 67-82.

[16] I. G. Macdonald, Schubert polynomials, in Surveys in combinatorics, 1991 (Guildford, 1991), 73-99, London Math. Soc. Lecture Note Ser., 166, Cambridge Univ. Press, Cambridge, 1991.

[17] T. Matsumura, Flagged Grothendieck polynomials, arXiv:1701.03561.

[18] P. J. McNamara, Factorial Grothendieck polynomials, Electron. J. Combin. 13 (2006), no. 1, Research Paper 71, 40 pp.

[19] K. Motegi and K. Sakai, Vertex models, TASEP and Grothendieck polynomials, J. Phys. A 46 (2013), no. 35, 355201, 26 pp.

[20] O. Pechenik and A. Yong, Equivariant K-theory of Grassmannians, Forum Math. Pi 5 (2017), e3, 128 pp.

[21] M. Wheeler and P. Zinn-Justin, Littlewood-Richardson coefficients for Grothendieck polynomials from integrability, arXiv:1607.02396.

[22] D. Yeliussizov, Duality and deformations of stable Grothendieck polynomials, J. Algebraic Combin. 45 (2017), no. 1, 295-344.

By Tomoo MATSUMURA

Department of Applied Mathematics, Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama 700-0005, Japan

(Communicated by Masaki KASHIWARA, M.J.A., Sept. 12, 2017)
COPYRIGHT 2017 The Japan Academy
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2017 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Matsumura, Tomoo
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
Article Type:Report
Date:Aug 1, 2017
Words:1750
Previous Article:A character of the Siegel modular group of level 2 from theta constants.
Next Article:Symmetry breaking operators for the restriction of representations of indefinite orthogonal groups O(p, q).
Topics:

Terms of use | Privacy policy | Copyright © 2021 Farlex, Inc. | Feedback | For webmasters |