Class Note for MATH 796 at KU 3
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Date Created: 02/06/15
Wednesday 423 Frobenius Reciprocity Let H C G be nite groups and let X be characters of G and H respectively The restricted character of 1 on H is 1 Resg Wh 00 and the induced character of X on G is 1 7 2 Indi y m E Xva 1976 keG k lgkeH Theorem 1 Frobenius Reciprocity lt1nd x gtG ltX Resg Proof 1 m C 2 XUVIQMWQ by 2 9amp0 keG k lgkeH 7 7 71 7 GHH Z xlthgtwltk 9k 1 7 7 71 i W 1679 khk 1 See Monday s notes for an application and there Will be more later Symmetric Functions De nition 1 Let R be a commutative ring typically Q or Z A symmetric function is a polynomial in Rzli i In that is invariant under permuting the variables For example if n 3 then up to scalar multiplication the only symmetric function of degree 1 in 111213 is 11 12 13 In degree 2 here are two I I I 7 11121113I2I Every other symmetric function that is homogeneous of degree 2 is a Rlinear combination of these two because the coef cients of I and 1112 determine the coef cients of all other monomialsi Note that the set of all degree2 symmetric functions forms a vector space ln degree 3 the following three polynomials form a basis for the space of symmetric functions 1 13 I 1 12 111 1 13 111 1313 121 111213r Each member of this basis is a sum of the monomials in a single orbit under the action of g Accordingly we call them monomial symmetric functions and index each by the partition whose parts are the exponents of one of its monomialsr That is 1311712713 Ii 13 137 m21111213 1 12 111 1 13 111 1313 121 m11111z27 Is 111213 In general for A A1 r r r Ag we de ne A A A m 11rrr1n Z 1ai1ajn1afr 0102Clnl But unfortunately this is zero if I gt n So we need more variables In fact we will in general work with an in nite set of variables 1112 r r r De nition 2 Let A b n The monomial symmetric function m is the power series A A A m Z 1ai1a 1afr a1azCP That is m is the sum of all monomials whose exponents are the parts of Ar Another way to write this is m E 10 rearrangements a of A where 10 is shorthand for 1 11 2 r Here we are regarding A as a countably in nite sequence in which all but nitely many terms are Or We then de ne Ad Aggy degreed symmetric functions with coefflts in R AARAdr 0120 Each Ad is a nitedimensional vector space with basis mk l A b d dimc Ad pd the number of partitions of d and the dimension does not change even if we zero out all but d variables so for many purposes it is permissible and less intimidating to regard Ad as the space of degreed symmetric functions in d variablesr Moreover A is a graded ring In fact let 600 be the group whose members are the permutations of 1112 r r with only nitely many non xed points that is 600 j 6 711 Then A Rllzl7127 7ll6w This understandably bothers some people In practice we rarely have to worry about more than nitely many variables when carrying out calculations Where is all this going The punchline is that we are going to construct an isomorphism F AQ 69 CZQGn n20 called the Frobenius characten39stic Thus will allow us to translate symmetric function identities into statements about representations and characters of Gm and Vice versa Important Families of Symmetric Functions Throughout this section7 let A A1 2 A2 2 2 M b n 1 Monomial symmetric functions These we have just seen 2 Elementary symmetric functions For k E N we de ne ek E HIS E IiIIiZHIik mu1 SCN 5E5 0lti1lti2ltmltik Sk where there are k 1 s in the last expression In particular 60 1 We then de ne E 6 Z For example7 1111I2Is2 I I 2I1121113I2131114quot39 m22m117 21 11I213393911121113I2131114quot39 m213m1117 11111I2133 ms 3m216m1117 et cetera Observe that 3 Et 2 HQ in Ztkek 13921 kgo 3 Complete homogeneous symmetric functions For k E N we de ne hk to be the sum of all monomials of degree 16 hk 2 H15 Z zilziznzik Em multisets SCN 5E5 1151535116 AHc k We then de ne hA hA1quot39hw For example7 hm 611 and h21 h1h2 e177111 m2 e1611 62 6111 i 621 ms 2m21 3min The analogue of 4 for the homogeneous symmetric functions is 4 Ht H 171m Ztkhki 13921 kgo In many situations7 the elementary and homogeneous symmetric functions behave dually As we Will see7 the sets EA l A F d and hA l A F d are Z module bases for Adi
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