Preface
These note s presen t som e application s o f th e theor y o f symmetri c function s
to algebrai c identitie s relate d t o th e Euclidean algorithm: divisio n o f polynomi -
als, continue d fractio n expansio n o f forma l series , Pad e approximants , orthogona l
polynomials.
Most o f the time , statement s wil l involve onl y Schur functions, speciall y thos e
indexed b y a rectangular partition , whic h shoul d b e considere d a s generalizing th e
resultant o f two polynomial s i n one variable .
Proofs rel y o n les s tha n hal f a doze n formula s wit h symmetri c functions , sup -
plemented b y tw o identitie s abou t minor s (Plucke r relation s an d Bazi n relations) .
Some statements ar e prove d severa l time s i n differen t manners , an d w e could hav e
even restricted ou r toolkit b y not usin g identities about minors . Bu t by doing so, we
would hav e misse d th e fac t tha t man y algebrai c construction s involvin g Hanke l o r
Toeplitz determinant s ar e closely related t o the theory o f symmetric functions , an d
that suc h a n interpla y i s a fruitfu l manne r o f avoidin g unnecessar y computations ,
and o f providing connection s betwee n differen t fields o f mathematics .
As an illustration , w e have take n man y example s fro m Muir' s five volumes the
Theory of Determinants in the Historical Order of Development [139], referrin g
to Muirl,... , MuirV . Man y mor e example s coul d hav e bee n added . Fo r example ,
to reformulat e Muir' s section s abou t alternants i n terms o f symmetric function s i s
straightforward.
Our las t chapter s sho w ho w t o adap t th e previou s construction s t o non-sym -
metric polynomials . Essentially , Schu r function s ar e replace d b y th e mor e genera l
family o f Schubert polynomials. T o introduce Schuber t int o the story , we first nee d
to forge t abou t determinants , an d kee p onl y th e underlyin g symmetri c groups .
Reducing t o &2 ? t o 2 x 2 determinants an d t o polynomial s i n two variables reveal s
itself t o b e the ke y fo r computin g i n severa l variables .
This metho d wa s initiate d b y Newton , wh o applie d i t t o obtai n a n interpola -
tion formul a whic h i s a discret e versio n o f Taylor' s formula , replacin g derivative s
by divided differences. W e sho w ho w t o obtai n a simila r interpolatio n i n severa l
variables, wit h Schuber t polynomial s occurrin g a s universal coefficients , I n the fol -
lowing chapter, w e abandon interpolatio n methods , an d rathe r sho w that Schuber t
polynomials ca n b e obtaine d throug h a Cauchy-typ e kernel , exactl y a s with Schu r
functions.
We have sometimes departed fro m th e usual conventions about symmetri c func -
tions, a s foun d i n th e treatis e o f Macdonald . Fo r example , Schu r function s ar e
indexed b y increasing partitions t o make it easier to use minors, or also, to conside r
them a s special Schuber t polynomials .
We als o us e A-ring s a s th e prope r framewor k fo r symmetri c functions . Th e
reader wil l se e tha t man y identitie s boi l dow n t o (A+B ) C = A + (B-C ) i n thi s
vii
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