### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Class Note for MATH 796 at KU 3

### View Full Document

## 12

## 0

## Popular in Course

## Popular in Department

This 4 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Kansas taught by a professor in Fall. Since its upload, it has received 12 views.

## Reviews for Class Note for MATH 796 at KU 3

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I made $350 in just two days after posting my first study guide."

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.