Introduction to symmetric functions

F om symme ic unc ions o qubi s F om symme ic unc ions o qubi s Ou pu pose: Explain how o compu e he Hilbe se ies o he algeb as o co a ian s o (pu e) qubi s sys ems. Why hese algeb as a e ele an ? The (pu e) qubi sys ems a e ega ded as mul ilinea o ms on which ac s a p oduc o linea g oups (SLOCC: S ochas ic Local Ope a ions and he Classical Communica ion). The knowledge o he co a ian s allows: - (In p inciple) o desc ibe he s uc u e o he o bi s. - Cons uc en anglemen mono ones and measu e o en anglemen . F om symme ic unc ions o qubi s Ou plan: Lec u e I: In oduc ion o symme ic unc ions (Jean-Gab iel Luque) We will p esen he main ool: he symme ic unc ions. Lec u e II: Symme ic unc ions and cha ac e s o he symme ic g oups.(JGL) We will explain why he symme ic unc ions encodes he cha ac e s o he symme ic g oups. Lec u e III: Ve ex ope a o and K onecke coe icien s (Jean-Y es Thibon) We will explain he links be ween he K onecke coe icien s and he symme ic unc ions. Lec u e IV: Hilbe se ies o he algeb as o in a ian s o a k-qubi s (pu e) sys em. (JYT) We will explain how o use hese ools o compu e he Hilbe se ies. F om symme ic unc ions o qubi s Lec u e I In oduc ion o symme ic unc ions Jean-Gab iel Luque Symme ic unc ions De ini ion: Polynomials in se e al a iables (alphabe X={x1,...,xn,...}) which a e in a ian unde pe mu a ions o he a iables. Sym(X) : algeb a o symme ic polynomials o he alphabe X Example The monomial unc ions: Symme ic unc ions De ini ion: Polynomials in se e al a iables (alphabe X={x1,...,xn,...} which a e in a ian unde pe mu a ions o he a iables. Sym(X) : algeb a o symme ic polynomials o he alphabe X Example The comple e unc ion hn is he sum o all he monomials o deg ee n Symme ic unc ions De ini ion: Polynomials in se e al a iables (alphabe X={x1,...,xn,...} which a e in a ian unde pe mu a ions o he a iables. Sym(X) : algeb a o symme ic polynomials o he alphabe X Example The elemen a y unc ions Symme ic unc ions De ini ion: Polynomials in se e al a iables (alphabe X={x1,...,xn,...} which a e in a ian s unde pe mu a ion i he a iables. Sym(X) : algeb a o symme ic polynomials o he alphabe X Example The powe sums Bases Suppose ha X={x1,...,xn,...} is an in ini e alphabe . Mul iplica i e bases: Non mul iplica i e bases: Comple e unc ions Powe sums Elemen a y unc ions Monomial unc ions Schu unc ions Di e ences o alphabe s P oduc o alphabe s A powe sum o a p oduc o alphabe s is he p oduc o he powe sums. Hence, Scala p oduc and ep oducing Ke nel (1) Conside a scala p oduc : Wi h a pai o bases in duali y The ep oducing ke nel associa ed o { , } is a mul i a ia e se ies on wo alphabe s de ined by Why his se ies is called a ep oducing ke nel? Scala p oduc and ep oducing Ke nel (2) As a consequence, i one has Fo any pai o bases in duali y: Usual scala p oduc and inne p oduc (1) We se Since, This is he ep oducing ke nel o he usual scala p oduc de ined by O equi alen ly, The usual inne p oduc is de ined by Usual scala p oduc and inne p oduc (2) W i e Since The coe icien o in is Mo e gene aly O hogonalisa ion o comple e unc ions One apply he G amm-Schmid o hogonalisa ion p ocess o he comple e unc ions o he in e se o de o dominance Example o he deg ee 4: ..... By de ini ion he a e o hogonal. This basis is called he Schu basis. An o hono mal basis (1): Semi-s anda d ableaux Semi-s anda d (Young) ableaux o shape We ill up he nodes o he shape wi h non-nega i e in ege s such ha he en ies a e s ic ly inc easing along each column and jus non-deg easing along each ow. no no ok An o hono mal basis (2): Schens ed algo i hm Schens ed algo i hm: cons uc a semi-s anda d ableau om a sequence o in ege s I e a ion o he inse ion: Inse ion o an in ege n in o a ableau 1) Fi s y o inse on he i s line. ...... ...... ...........m n I m< n he esul is ob ain by glueing n a he end o he line 2) o he wise, le p he smalles in ege o he line s ic ly g ea e ha n ........... ........... .......p........m n Replace he i s occu ence o p by n and y o add p in o he nex line ........... ........... p .......n........m An o hono mal basis (3): Schens ed algo i hm, example Conside he sequence 42214163311 4 4 42 2 4 22 4 4 42 2 221 12 12 4 2 124 4 4 2 2 2 1241 1 1 4 4 22 1146 4 4 22 224 11463 1136 4 4 224 2246 11363 1133 4 4 44 2246 22463 2236 11331 1113 1113 44 44 446 2236 22363 2233 11131 1111 1111 Schu o m Hence whe e a e he Kos ka numbe s Example: h o Schu Since a e in duali y The Schu basis being o hono mal, one has implies Conclusion + We ha e shown how o use gene a ing unc ion o + Compu ing changes o bases. + De ining he scala s anda d scala p oduc and he s anda d inne p oduc . + We ha e de ined Schu unc ion by o hogonalizing he comple e unc ions. + We ha e p o ed ha he Schu unc ions a e o hono mal o he s anda d scala p oduc .