IMPA Monographs Volume 4

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Letterio Gatto • Parham Salehyan

Hasse-Schmidt Derivations on Grassmann Algebras With Applications to Vertex Operators

123

Letterio Gatto Politecnico di Torino Torino, Italy

Parham Salehyan São Paulo State University São José do Rio Preto Campus São José do Rio Preto, Brazil

IMPA Monographs ISBN 978-3-319-31841-7 ISBN 978-3-319-31842-4 (eBook) DOI 10.1007/978-3-319-31842-4 Library of Congress Control Number: 2016936657 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

A Sheila e ai nostri 10 figli, Giuseppe e Giuliano be m¯adar va pedar-e azizam

Preface

This book is a revised and expanded version of the material that the first author presented in a mini-course given at IMPA, about the same topics, on the occasion of the 30ı Colóquio Brasileiro de Matemática (July 2015), and is largely based on the collaboration with the second author and Inna Scherbak. It aims at introducing and advertising the notion of Hasse–Schmidt derivation on a Grassmann algebra, which we propose as a natural language to put many seemingly unrelated subjects into a unified framework. The broadened perspective it offers is able to capture, for example, a generalization of the Cayley–Hamilton theorem, the theory of linear ODEs and their generalized Wronskians, the exponential of a matrix with indeterminate entries (another reading of Putzer’s method dating back 1966), the characterization of decomposable tensors in an exterior power of a free abelian group and the bosonic expressions of the vertex operators generating the so-called free fermionic vertex superalgebra. The latter comes from the representation theory of the oscillator Heisenberg algebra, which is one of the fancy terms scattered throughout the text, other such being bosonic and fermionic Fock spaces, boson–fermion correspondence, vertex operators, vertex algebra and KP hierarchy, to name a few. We believe that the common formalism underpinning the aforementioned subjects proves interesting for its own sake. At the same time we wish to stress, once and for all, that this book is about none of them. Readers are not assumed to have any prior familiarity with such a language, for which we point out throughout the text many existing and excellent references. The book is suited for Ph.D. students wishing to learn additional features of exterior algebras in connection with more advanced topics. It is our hope, however, that diverse scholars, both algebraic geometers and not, will profit from it:we tried

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Preface

to illustrate material that may not belong to everybody’s field of research – or taste – but at the same time has the subtle potential of appealing to many, due to its farreaching breadth. Torino, Italy São José do Rio Preto, Brazil January 2016

Letterio Gatto Parham Salehyan

Acknowledgments

First and foremost, we acknowledge the organizing committee of the 30ı Colóquio Brasileiro de Matemática (IMPA, 27–31 July 2015) for hosting the short course that led to these notes, notably professor Carolina Araujo to whom our deepest feelings of gratitude go: her support was invaluable and eventually instilled in us the necessary enthusiasm to take on this project. We are also deeply indebted to professor Paulo Sad, for having encouraged us to submit the present text for publication within this monograph series, and to Simon G. Chiossi, who generously and patiently read all the preliminary drafts, making abundant remarks here and there and enabling us to substantially improve the exposition, not to speak of his truly invigorating support. Extra special thanks are also due to professors Marcos Jardim and Paolo Piccione for unreserved support. The first-named author (LG) is indebted to Inna Scherbak, for her enduring and friendly collaboration and for her many remarks on the first version of this work; to Maxim Kazarian, who first suggested the relationship between derivations on an exterior algebra and the boson–fermion correspondence; to Ilka Agricola, Simon Chiossi, Martin Gulbrandsen and Andrea Ricolfi for hosting preliminary talks on the subject in Marburg and Stavanger; and especially to Jorge Cordovez for having been very close to us during the redaction process and to Maria Jack for language consultancy. Moreover, LG is deeply thankful to Anders Thorup and Gary Kennedy for long-standing friendship; to Alex Kasman, Igor Mencattini and Carlos Rito for stimulating discussions on PDEs; to Alessandro Ardizzoni and Paolo Saracco for enlightening explanations on infinite exterior powers; and to professor Louis Rowen for precious suggestions, not to mention his tenacious backing. Both authors are grateful to the head of the Scientific Office at the Italian Embassy in Brasilia, Professor Roberto Bruno, for his personal and institutional support. For many diverse reasons, we are also thankful to Roberto Alvarenga, Dan Avritzer, Claudio Bartocci, Roberto Bedregal, Pablo Braz e Silva, Ugo Bruzzo, Frediano Checchinato, Valeria Chiadò-Piat, Marco Codegone, Giovanni Colombo, Marc Coppens, Fernando Cukierman, Caterina Cumino, Eduardo Esteves, Fabio

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Acknowledgments

Favaro, Umberto Ferri, Anna Fino, Paolo Giaquinta, Flavia Girardi, Sabrina Gonzatto, Antonella Grassi, Abramo Hefez, Joachim Kock, Daniel Levcovitz, Angelo Felice Lopez, Angelo Luvison, Simone Marchesi, Marina Marchisio, Luisa Mazzi, Nivaldo Medeiros, Paolo Mulassano, Barbara Mussio, Antonio Nigro, Karl–Otto Stöhr, Marco Pacini, Diego Parentela, Elisa Pilot, Pietro Pirola, Piotr Pragacz, Francesco Raffa, Gaetana Restuccia, Paulo Ribenboim, Andrea Ricolfi, Clara Silvia Roero, Lucio Sartori, Boris Shapiro, Aron Simis, Fernando Xavier Souza, Carlos Tomei, Mario Trigiante, Emanuela Truzzi, Israel Vainsencher and, especially, Laura Tonini, who helped LG to look at the world with different eyes. Moreover, both authors are deeply grateful to Robinson Nelson dos Santos of the Springer Editorial Staff for careful editorial assistance and Mr. Santhamurthy Ramamoorthy of the Spi Global, for his very precious help to improve the very final version of the notes, as well as the Production Editor Mr. Anthony Reagan Chinappan. Last, but certainly not least, we owe a lot to the precious and friendly help of the entire staff at IMPA, notably Maria Celano Maia, Priscilla Fernandes Pomateli, Rogério Dias Trindade, Rafael Simão Rodriguez, Noni Geiger, Ana Paula da Fonseca Rodrigues and Sergio Ricardo Vaz, as well as Suely Lima, Pedro Luis Darrigue de Faro, Sonia M. Alves, Juliana C. Bressan, Leticia R. Nascimento, Jurandira F. R. Nascimento, Paula Cristina R. Dugin, Rosana de Souza, Vanda Silvestre and Roseni Victoriano. This work was partially supported by: Instituto Nacional de Matemática Pura e Aplicada (IMPA); Italy’s ‘National Group for Algebraic and Geometric Structures, and their Applications’ (GNSAGA–INdAM); FAPESP-Brazil, Proc. 2012/028691 and 2015/04513-8, Ufficio Scientifico dell’Ambasciata d’Italia a Brasilia; PRIN ‘Geometria delle varietà algebriche’; UNESP – Campus de São José do Rio Preto; Dipartimento di Scienze Matematiche ‘G. L. Lagrange’ del Politecnico di Torino; Istituto Superiore Mario Boella and Madi; Medas Solutions and Filters srl. All these institutions are warmly acknowledged.

Partial List of Notation

We adopt general standard conventions: Z; N; Q; C denote, respectively, the ring of the integers, the monoid of non-negative integers and the field of the rationals and of the complex numbers. We also use the notation ZC D Z0 D N. The positive integers are denoted by N . The unit of a ring A is denoted 1A or simply 1 if clear from the context. The identity endomorphism of an A-module M is denoted by 1M . MŒŒt AŒX

Formal power series with M-coefficients . . . . . . . . . . . . . . . . . . . . . . . . . The ring of polynomials in an indeterminate X with coefficients in A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M..t// Formal Laurent series MŒt1 ; t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . det./ Determinant of a square matrix or of an endomorphism . . . . . . . . . . . The polynomial ring ZŒe1 ; : : : ; er  in r weighted Br indeterminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The submodule of Br of elements of weight w  0 . . . . . . . . . . . . . . . . .Br /w Generic monic polynomial of degree r . . . . . . . . . . . . . . . . . . . . . . . . . . . pr .X/ P and Pr The set of all partitions and of the partition of length at most r . . . . . . The set of partitions of Pr bounded by n-r . . . . . . . . . . . . . . . . . . . . . . . . Pr;n j .............................................................. D The Schur polynomial associated to a partition  and to a  .a/ sequence a in a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The polynomial 1  e1 t C    C .1/r er tr 2 Br Œt . . . . . . . . . . . . . . . . . Er .t/ P The inverse n2Z hn tn of Er .t/ in the ring Br ŒŒt . . . . . . . . . . . . . . . . . . Hr .t/ The sequence .hj /i2Z of the coefficients of Hr .t/ . . . . . . . . . . . . . . . . . . Hr The fundamental sequence of Br ŒŒt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ui /i2Z .............................................................. Ui

24 24 25 25 26 27 26 27 27 26 28 28 29 29 30 31

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Partial List of Notation

Residue of a formal Laurent series f 2 AŒX 1 ; X, the coefficient of X 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kr ; Kr .A/ and Kr .M/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................................ pr .D/ The formal derivative on the algebra AŒŒt . . . . . . . . . . . . . . . . . . @t L The formal Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . F The r-tuple .f1 ; : : : ; fr / of polynomials of AŒX . . . . . . . . . . . . . . The r-tuple of polynomials .X riC1 Ci1 /1ir . . . . . . . . . . . . . . X The canonical basis .u0 ; u1 ; : : : ; urC1 / of Kr . . . . . . . . . . . . . . ur F-Wronskian associated to ur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WF .ur / The Wronskian associated to r-tuple X . . . . . . . . . . . . . . . . . . . . WX .ur / Tensor algebra of an A-module M . . . . . . . . . . . . . . . . . . . . . . . . . . TA .M/ V M Exterior (or Grassmann) algebra of an A-module M . . . . . . . . . . r ........................................................ Œbi ........................................................ ŒbriC Cg .M/ Clifford algebra associated to a symmetric bilinear form g on a free abelian group M . . . . . . . . . . . . . . . . . . . . . . . . . . exp.t/ The exponential formal power series . . . . . . . . . . . . . . . . . . . . . . . y Contraction operator associated to  2 M _ . . . . . . . . . . . . . . . . . V V V MŒŒz The algebra . M/ŒŒz of M-valued formal power series . . V The unit of the A-algebra M (coinciding with 1A ) . . . . . . . . . 1V M V The identity endomorphism of M . . . . . . . . . . . . . . . . . . . . . . . . 1V M The identity endomorphism of M . . . . . . . . . . . . . . . . . . . . . . . . . . 1M ........................................................ evC C .z/;  C .z/ Schubert derivations (one inverse of the other) . . . . . . . . . . . . . . ........................................................ Mp The complex Grassmann variety of r-planes in Cn . . . . . . . . . . . Gr;n The Schubert cycle associated to the partition  . . . . . . . . . . . . .  A free abelian group of infinite countable rank M0 (e.g. ZŒX) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A free abelian group of finite rank n (e.g. ZŒX=.X n /) . . . . . . . . M0;n R  Gr;n c \ ŒGr;n  The degree of a Chow class c 2 A .Gr;n / . . . . . . . . . . . . . . . . . . . Vr The evaluation morphism Br ! M0 . . . . . . . . . . . . . . . . . . . . . evŒbr0 V Br;n A quotient of Br , isomorphic to r M0;n . . . . . . . . . . . . . . . . . . . . The dual module of M0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M0_  The restricted dual of the module M0 . . . . . . . . . . . . . . . . . . . . . . . M0 The Z-linear span of .ui /i2Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ur ResX f

30 32 32 36 36 39 40 39 39 40 52 52 55 55 55 48 60 63 63 64 68 84 82 82 90 93 105 105 98 106 112 122 123 148

Partial List of Notation

U? r ŒuriC ˆri ˆriC G . C / G .Hr / n0 nr  .z/ and   .z/ ˇ.z1 / B1 Fir r

F W.ur / jiir Hr @i Br b DC .z/; DC .z/ D .z/; D .z/ Br .`/ Xr .z/; Xr .z/

r .z/

r .z/

.z/ and  .z/ R.z/

...................................................... ...................................................... ...................................................... ...................................................... ...................................................... Giambelli polynomial (2 Br ) corresponding to Œbr . . . . . . . ...................................................... ...................................................... Schubert derivations (one inverse of the other) . . . . . . . . . . . . P The formal power series j0 ˇj zj1 . . . . . . . . . . . . . . . . . . . The polynomial ring ZŒe1 ; e2 ; : : : in infinitely many indeterminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fermionic Fock space of order r and charge i . . . . . . . . . The fermionic Fock space of order r . . . . . . . . . . . . . . . . . . . . . The Wronskian module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The vacuum vector of charge i of order r . . . . . . . . . . . . . . . . The .2r C 2/-dimensional Heisenberg algebra . . . . . . . . . . . . @ The partial derivative ............................... @xi The ring Br ˝Z Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trace of a Q-endomorphism of Br ˝Z Ur . . . . . . . . . . . . . . . . . The Schubert derivations, one inverse of the other, V defined on Ur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Schubert derivations, one inverse of the other, V defined on Ur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong abuse of notation. It stands for Br Œ`1 ; ` . . . . . . . . . . Fermionic vertex operators truncated to the order r . . . . . . . Bosonic counterpart of Xr .z/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bosonic counterpart of Xr .z/ . . . . . . . . . . . . . . . . . . . . . . . . . . .  A shorthand for 1 .z/ and 1 .z/, respectively . . . . . . . . . . . An operator Br .`/ ! Br .`/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

148 149 149 149 106 107 111 111 125 133 142 149 149 151 152 153 154 153 155 161 161 168 169 169 169 177 177

Introduction

As the title gives away, this book is concerned with Hasse–Schmidt (HS) derivations on Grassmann (or exterior) algebras. These were proposed in [37] as an alternative formalism for dealing with Schubert calculus on complex Grassmannians. They are a straightforward extension of a corresponding notion [60] arising in commutative algebra. The guiding expository strategy with these notes is to exploit HS derivations to dig up a number of lands in the realm of elementary linear algebra to uncover the roots of more advanced topics. • HS-derivations, in general. To begin with, let A be a commutative ring with unit and .M; ?/ an associative A-algebra. An P M-valued formal power series in an indeterminate z over M is an infinite sum i0 mi zi encoding the data of an Mvalued sequence .m0 ; m1 ; : : :/. Denote by MŒŒz the set of all M-valued formal power series, endowed with its standard A-algebra structure. Extending the terminology of [112, p. 207], by an HS-derivation on .M; ?/ we shall understand an algebra homomorphism D.z/ W M ! MŒŒz, i.e. an A-linear map such that D.z/.m1 ? m2 / D D.z/m1 ? D.z/m2 for all m1 ; m2 2 M. Equivalently, D.z/ is a formal power series whose underlying sequence .D0 ; D1 ; : : :/ of endomorphisms of M obeys the following higher-order Leibniz-like rule Dn .m1 ? m2 / D

i X

Dnj m1 ? Dj m2 ;

.i  0/:

(1)

jD0

• HS-derivations: à la Hasse and Schmidt. If .M; ?/ happens to be a commutative A-algebra, the definition sketched above is the same that Hasse and Schmidt introduced back in 1937, under the name of higher derivation [60]. The purpose of those authors, mainly interested in arithmetic questions, was to build a characteristic-free algebraic analogue of the Taylor series of a smooth function. Two years later, Schmidt constructed Wronskian determinants associated to higher derivations in order to study Weierstrass points on curves in positive

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Introduction

characteristic [137]. Ever since, the literature concerning HS-derivations in the commutative-algebra setting has grown relentlessly. Besides the erstwhile aforementioned work of Hasse and Schmidt and the standard reference [112, Ch. 4], it is dutiful to cite the papers by Ribenboim [127–130], Saymeh [135], Laksov-Thorup [95, Section 4], Vojta [146] and the more recent works by Esteves and Skjelnes [27, 142] as an indication of the vitality of the subject. • HS-derivations: on a Grassmann algebra. Surprisingly, the literature appears toVmiss, if not ignore completely, the case where .M; ?/ is the exterior algebra . M; ^/ of a free A-module M. The main purpose of this V book is to fill such gap. We shall do so by defining an HS-derivation on M as an algebra homomorphism D.z/ W

^

M!

^

MŒŒz;

(2)

V V V where MŒŒz WD . M/ŒŒz denotes the algebra of M-valued formal power series. That is nothing fancy, of course, as it merely amounts to clone p. 207 of Matsumura’s book of AV [112]. It turns out that the Punique sequence .Dn /n0 V endomorphisms of M defined by D.z/ D n0 Dn   zn , for all  2 M, satisfies the relation Dn . ^ / D

n X

Dnj  ^ Dj ;

(3)

jD0

which is exactly formula (1) up to replacing ‘?’ with the wedge product ‘^’. Formula (3) is obtained by comparing the coefficients of zn on either side of the equality D.z/. ^ / D D.z/ ^ D.z/ ; which holds by definition, as in the commutative case. It follows that D0 is a homomorphism of A-algebras V (D0 . ^ / D D0  ^ D0 ), while D1 is an A-derivation of M, that is, besides being A-linear, it satisfies the ordinary Leibniz rule D1 .^ / D D1 ^ C^D1 . This passing comparison with the classical commutative case had a purpose: to convince V V the reader that calling HS-derivation an algebra homomorphism M ! MŒŒz rightly answers the need for a truly appropriate terminology. • Integration V by parts. If an HS-derivation is additionally invertible, thought V of as an EndA . M/-valued formal power series, its inverse D.z/ 2 EndA . M/ŒŒz turns out to be an HS-derivation too. This seemingly harmless remark provides us with the integration by parts formula D.z/ ^ D D.z/. ^ D.z/ / ;

(4)

a rather obvious equality that, on the other hand, is probably the most powerful tool of the theory. Formula (4) will be applied throughout the book to a wide variety of situations: to name a few, a useful and major generalization [51] of the known Cayley–Hamilton theorem, whereby any endomorphism is a root of its own characteristic polynomial, Giambelli’s formula for the Schubert derivations

Introduction

xvii

treated in Chapter 5 and the approximate expressions of the vertex operators occurring in the representation theory of the oscillator Heisenberg algebra. To sense more intensely the flavour of the kind of results we shall depict within our framework, an especially relevant situation is described below. • Decomposable tensors in an exterior power. Adopting the same convention of Chapter 5, let us denote by M0 the free abelian group of infinite countable rank with basis .b0 ; b1 ; : : :/. An example of such a module is the additive group ZŒX of polynomials with integral coefficients. The reader who feels a little unfamiliar with the notion of exterior algebra and exterior V0 power of M0 is referred to Section 3.2. However, recall that by definition M0 D Z and that, for ˇr  1, the V rth exterior power r M0 is the free abelian group generated by .Œbr ˇ  2 Pr /, where Pr is the set of finite sequences .1 ; : : : ; r / 2 Nr such that 1  : : :  r , and Œbr WD b0Cr ^ b1Cr1 ^    ^ br1C1 : P V In particular, each  2 r M0 can be uniquely expressed as 2Pr;n a Œbr , where Pr;n WD f 2 Pr j 1  n  rgVis the set of partitions of length at most r bounded by n  r. An element  2 r M0 is said to be decomposable if  D m1 ^    ^ mr for some m1 ; : : : ; mr 2 M0 .V The issue is to determine equations for the locus of decomposable tensors in r M0 . The solution to this classical problem is very well known (see, e.g. [55, 58, 64]), but we shall revisit it here as follows. First recall that if a WD .ai /i2Z is a bilateral sequence of elements in a commutative ring, the Schur determinant associated to ‘a’ and  2 Pr is  .a/ WD det.aj jCi /1i;jr : Let B0 WD Z and, for r  1, define Br WD ZŒe1 ; : : : ; er  to be the polynomial ring in the r indeterminates .e1 ; : : : ; er /. The reason for the notation is that the complexification B WD B1 ˝Z C coincides with the bosonic (whence the ‘B’) Fock representation of the Heisenberg oscillator algebra, i.e. the Weyl affinization [73, p. 51] of the trivial Lie algebra C (cf. Remark 1.2.3). Accordingly, for any r  0, we define polynomials Er .z/ 2 Br Œz as follows: E0 D 1 and Er .z/ WD 1  e1 z C    C .1/r er zr for r > 0, in order to construct the unique Br -valued sequence Hr WD .hi /2Z such that X i2Z

hi zi D

1 ; Er .z/

where the equality is understood in the abelian group of formal Laurent series Br ŒŒz1 ; z. In particular, h0 D 1 and hj D 0 if j < 0. Toe each hn 2 Br we associate the following two Laurent polynomials   .z/hn WD hn  hn1 z1

and

 .z/hn WD

X i0

hni zi :

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Introduction

Denote by   .z/Hr (resp.  .z/Hr ) the sequence .  .z/hi /i2Z (resp. . .z/hi /i2Z ). Then, the main result of Section 6, Theorem 6.4.3, guarantees that the linear combination X a Œbr (5) 2Pr;n

V is decomposable in r M0 if and only if the integers a satisfy the following system of bilinear equations:

Resz

X

;2Pr

a a Er1 .z/  . .z/Hr1 / ˝

1

 .  .z/HrC1 / D 0; ErC1 .z/

(6)

where ‘Resz ’ denotes the residue of a Laurent series, i.e. the coefficient of the monomial z1 . It is dutiful to remark that, in spite of its elegance, formula (6) is not very efficient from a computational point of view. Examples 6.4.6 and 6.4.8 show that detecting decomposable tensors already in the second wedge power of a module V of rank 4 or 5 is quite painful computationally, while the condition ^ D 0 for  2 2 M0 gives the result in a few instants (Exercise 5.11.1). There are a number of features, however, that make formula (6) interesting, shortly listed below: 1) It implicitly involves, through the formal power series 1=ErC1 .z/ V and the polynomial Er1 .z/, the data of two distinguished HS-derivations on M0 . They are denoted by C .z/ and  C .z/, respectively, and called Schubert derivations in compliance with the terminology introduced in [49]. The former is characterized P i by the equality  .z/b D b C j i0 iCj z 2 M0 ŒŒz, for all j  0, while the latter is V its inverse in M0 ŒŒz. 2) For each r > 0, the Schubert derivations enable to turn the basis .b0 ; b1 ; : : :/ of M0 into a generic linear recurrence sequence of order r (Chapter 2), which endows M0 with a V structure of free Br -module of rank r, denoted by Mr . V Decomposability in r M0 is then read in r Mr , a free Br -module of rank one generated by b0 ^ : : : ^ br1 . This is crucial for simplifying the algebraic manipulations leading to (6). 3) V Formula (6) renders explicit the fact that detecting decomposable V tensors on r r1 Mr amounts to exploit the interaction between the B -module Mr1 r1 VrC1 and the BrC1 -module MrC1 . For r D 1, they become infinite exterior powers coinciding over B1 , and the relationship turns into an interaction between fermionic Fock spaces of different charges. 4) Chapter 5 will show that formula (6) is manifestly related with Schubert calculus, as it is built out of four fundamental blocks only: the Schubert derivations C .z/,  C .z/ and two novel HS-derivations, denoted by  .z/ and   .z/, respectively. The latter appear as a mirror of sorts of C .z/ and  C .z/ but, in contrast to them,

Introduction

xix

enjoy a nice stability property enabling to describe their action directly on the ring Br , as explained in Section 6. V 5) Classical Plücker equations of the Grassmann cone of r M0 are uniformly encoded in just one formula. Last, but not least, taking the limit for r going to infinity, and extending the coefficient ring to B WD B1 ˝ Q, equation (6) recovers precisely the celebrated KP hierarchy, so named after Kadomtsev and Petviashvili, from the theory of infinite-dimensional integrable systems. Or, as Sato’s Japanese School puts it, it provides the Plücker embedding of the universal Grassmannian manifold (UGM) parametrizing infinite-dimensional subspaces of an infinite-dimensional vector space. Having said all that, it is perhaps time to pass to the contents of the single chapters, as a teaser of what is to come. • Chapter 1 is a prelude of sorts, mainly to furnish motivations and not strictly essential for grasping the rest. It may be read casually, glossing over details that we made no attempt to clarify at this juncture. It begins by describing a few hidden algebraic peculiarities, unveiled by the work of many authors, of two renowned non-linear PDEs, the Korteweg–deVries equation (KdV) and the Kadomtsev–Petviashvili equation (KP). The so-called boson–fermion correspondence is also anticipated in this chapter, and Vertex operators generating the free fermionic vertex superalgebra, in the sense of [32, Ch. 5], are outlined in Section 1.3.3. They act on a polynomial ring in infinitely many indeterminates via the boson–fermion correspondence and encode the full KP hierarchy, a certain infinite system of non-linear PDEs, which in turn includes the KP equation itself. Due to remarkable work by Sato [133] (see [134] for an illuminating survey) and Date–Jimbo–Kashiwara–Miwa [18, 19, 70], the KP hierarchy can be amazingly interpreted as the infinite set of Plücker equations of the Grassmann cone of decomposable tensors in an infinite wedge power. This closes the circle on one hand and paves the way to Chapter 6 on the other. • Chapter 2 presents generic linear recurrence sequences (LRS) with values in a module over a Br -algebra. Section 2.1 deals with LRSs viewed as formal power series, while Section 2.2 describes the weight graduation of Br using partitions. Section 2.5 discusses the formal Laplace transform and initial-value problems for linear ODEs. We work out two examples taken from [42] to explain how the method works. The reward for developing all this is the universal expression for solutions to Cauchy problems of linear ODEs with constant coefficients and analytic forcing terms [49, 50]. The formula arises from a distinguished sequence .ui /i2Z of Br -valued generic LRSs, for i  r C 1, introduced in [49] and generalized in [44]. They will resurface in Chapter 7, when their elementary properties will be used to determine the explicit expression of the vertex operators. • The first two sections of Chapter 3 collect basic algebraic facts that belong to any scholar’s background. In view of having a self-contained exposition, Section 3.1 opens with the notion of tensor algebra of a module, of which the exterior algebra

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is a quotient. The purpose of these preliminaries is to prepare the reader with the construction of the less common Clifford algebra of a module, as we do in Section 3.3. The infinite exterior power of an infinite dimensional vector space can be seen as a representation of a suitable canonical Clifford algebra. The concept of derivation on a Grassmann algebra, in the standard sense of derivations on any algebra, is reviewed in Section 3.4 to emphasize that iterated derivations yield higher-order Leibniz rules which can be themselves organized into an HS-derivation. • The pivotal notion of Hasse–Schmidt derivation on an exterior algebra is finally introduced in Chapter 4. Its basic formalism is developed, and the key role of the integration by parts formula (4) is discussed at length. HS-derivations on exterior algebras also provide a natural framework to state and prove, as in Section 4.2, a generalization, based on [51], of the classical celebrated Cayley– Hamilton theorem: any endomorphism is a root of its characteristic polynomial. It will be employed in Chapter 5 to equip a free abelian group of rank n 2 N[f1g with a structure of free Br -module of rank r  n. Section 4.3 offers a few simple, but noteworthy, applications to matrix exponentials and first integrals of linear ODEs with constant coefficients, adopting the same language of Section 2.5. • Chapter 5 is concerned with the Schubert derivations C .z/;  C .z/, introduced with a different notation in the papers [14, 37, 47] to deal with Schubert calculus for complex Grassmannians and recently used in [15] for application to the existence of certain indecomposable vector bundles. The  in the notation is reminiscent of its relationship with Schubert calculus, while the subscript ‘C’ keeps into account its kinship with vertex operators. The bridge connecting the Schubert derivation to classical Schubert calculus for Grassmannians, as summarized in Section 5.4, rests on a Pieri-like formula (Theorem 5.2.2) and a Giambelli formula satisfied by the coefficients of C .z/. The latter is proven by invoking an elegant and flexible determinantal formula due to Laksov and Thorup [96, 97] that will be also used, along the way, to construct approximations of the vertex operators generating the fermionic vertex superalgebra. Section 5.5 is entirely devoted to examples of manipulations with Schubert derivations carrying enumerative geometrical interpretations, such as computing degrees of Schubert varieties. • The main point of Chapter 6 is that the Schubert derivation C .z/, dealt with in Chapter 5, actually tells us one half of the story only. In fact, as anticipated in the first part of this P introduction, V there exists V a sort of ‘mirror’ Schubert derivation,  .z/ WD i0 i zi W M0 ! M0 ŒŒz1 , which is the unique HS-derivation such that j bi 7! bij if i  j and 0 otherwise (Section 6.2). The definition of  .z/ and its inverse   .z/ may seem rather artificial at first sight. However, they arise spontaneously from the theory. The very definition of C .z/ and integration by parts (4) imply the equality X i0

bi zi ^  D C .z/b0 ^  D C .z/.b0 ^  C .z//:

(7)

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The fact that, by Theorem 6.2.6, the leftmost term of (7) satisfies the equality C .z/.b0 ^  C .z// D zr C .z/  .z/. r  ^ b0 /; V in rC1 M0 ..z//, shows at a glance how C .z/ naturally carries  .z/ with it and the kinship of both with the vertex operators. The stability properties enjoyed by  .z/ and   .z/ enable to define them directly in terms of Z-module homomorphisms Br ! Br Œz1  as in (5), which in turn allow to describe the decomposable locus in exterior powers as we explained at the beginning of this introduction. • Chapter 7 is another take on the same material of Chapter 6, albeit from a more concrete point of view, due to the identification of M0 WD ZŒX with the Z-module spanned by generic LRSs of finite order. A basis of the latter is given by the fundamental sequence of formal power series .ui / introduced and discussed in Chapter 2. A formalism involving infinite exterior powers is explicitly used in this chapter, which is mainly devoted to the deduction of the bosonic expression of vertex operators arising in the fermionic vertex superalgebra, mentioned in the Prologue (Chapter 1). Among other things, we shall touch upon the bosonic and fermionic representation of the finite dimensional Heisenberg algebra, as well as the extension of the HS-derivation to the fermionic spaces of order r, in the sense of Section 7.6. Every chapter ends with one section about the related literature and another with exercises meant both as a complement to the subject and to highlight the connections with other topics that are not addressed directly in the text. Exercises range from routine ones, or examples that the authors worked out early on, to more involved that might offer novel insights into the theory. We should add, to conclude, that space and time constraints obliged us to leave out of the general picture many interesting and deserving aspects. In particular, we would have liked to include the equivariant cohomology of Grassmannians [48] and revisit Laksov’s two papers [93, 94] as well. We disregarded Chern classes of vector bundles defined in terms of trace polynomial operators (though they appear in an exercise) and left out the description of the notable Kempf–Laksov formula [84] in terms of the exterior algebra of a module over the cohomology ring of the base of the bundle. Nor have we touched the important and fascinating relationship with the celebrated Jacobi triple product identity, of which the boson–fermion is a categorification in the sense of [31]. This book was written in the attempt to be as interdisciplinary as possible. This is reflected in the assortment of areas touched upon: from algebra to differential equations, from geometry to mathematical physics. Since the level is more elementary than that of the aforementioned general references, no prerequisites beyond standard linear/multilinear algebra are needed. In fact, starting from Chapter 2, the text is entirely self-contained. An effort was made to render the presentation seamless and avoid sudden jumps, to the point that even well-known concepts such as tensor algebras are defined from scratch.

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Corrections, additional material, solutions of selected exercises and updates will be posted at the url: http://calvino.polito.it/~gatto/impa_monograph.htm Please, visit it now and consult it frequently.

Contents

1

Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The KdV and the KP Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Vertex Operators and Affine Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Vertex Operators and Vertex Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The KP Hierarchy via Vertex Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 11 14 17 21

2

Generic Linear Recurrence Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Sequences in Modules over Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Generic Polynomials, Partitions and Schur Determinants . . . . . . . . . . . 2.3 Generic Linear Recurrence Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Cauchy Problems for Linear Recurrence Sequences. . . . . . . . . . . . . . . . . 2.5 Formal Laplace Transform and Linear ODEs . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Generalized Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 24 26 30 34 36 39 45 46

3

Algebras and Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Tensor and Exterior Algebra of a Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Exterior Algebra of a Free A-Module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Exterior Algebras Versus Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Derivations, in General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Derivations on an Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 54 55 57 59 61 62

4

Hasse–Schmidt Derivations on Exterior Algebras . . . . . . . . . . . . . . . . . . . . . . . 4.1 Main Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Trace Operator Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Exponential of an Endomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 68 74 77 79 xxiii

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5

Schubert Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Generalities on Schubert Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Pieri Formula for Schubert Derivations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Grassmannian and Its Plücker Embedding . . . . . . . . . . . . . . . . . . . . . . 5.4 Relationship with Schubert Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Module Structures Induced by Schubert Derivations, I . . . . . . . . . . . . . . 5.7 Module Structures Induced by Schubert Derivations, II . . . . . . . . . . . . . 5.8 Giambelli Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Application to Modules of Finite Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 82 86 90 91 99 105 108 110 111 115 118

6

Decomposable Tensors in Exterior Powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 A Criterion for Decomposability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A Schubert Derivation with a Stability Property . . . . . . . . . . . . . . . . . . . . . 6.3 On Vertex-Like Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Plücker Equations for Grassmann Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 On the Infinite Exterior Power I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 122 125 132 137 142 143 144

7

Vertex Operators via Generic LRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Bosonic and Fermionic Fock Spaces of Finite Order . . . . . . . . . . . . . . . 7.2 The Finite Order Boson–Fermion Correspondence . . . . . . . . . . . . . . . . . . 7.3 On the Infinite Exterior Power II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 On the Finite-Dimensional Heisenberg Algebra . . . . . . . . . . . . . . . . . . . . . 7.5 Schubert Derivations on the Fermionic Modules . . . . . . . . . . . . . . . . . . . . 7.6 Extending the Boson–Fermion Correspondence . . . . . . . . . . . . . . . . . . . . . 7.7 Computing Truncated Vertex Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Vertex Operators in the Classical Boson–Fermion Correspondence 7.9 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 148 150 152 153 161 168 170 175 178 180

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193