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# Abstract Algebra MATH 307

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This 20 page Class Notes was uploaded by Miss Hillary Grady on Thursday October 29, 2015. The Class Notes belongs to MATH 307 at College of William and Mary taught by Christopher Vinroot in Fall. Since its upload, it has received 10 views. For similar materials see /class/231139/math-307-college-of-william-and-mary in Mathematics (M) at College of William and Mary.

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Date Created: 10/29/15

Several facts regarding U Math 307 Fall 2009 In class we de ned the group The de nition of Un was based on a Homework problem in a crucial way Here is the problem and its solution Problem Let a and n be positive integers and let d gcdan Prove that as E 1mod n has a solution if and only if d 1 Solution First assume that d 1 Then by the Corollary to Theorem 02 in the text which we proved in class there exist integers s and t such that as at 1 We may rewrite this equation as as 7 1 itn which means that n divides as 7 1 This implies by de nition or by the books de nition and by Exercise 9 on pg 22 that as E 1mod Therefore we have x s is a solution to ax E 1mod Conversely suppose that there is some solution x satisfying ax E 1mod Then 71 divides asi 1 and so there is an integer q such that ax 71 qn We may rewrite this equation as ax n7q 1 where z and 7q are integers By the last statement of Theorem 02 in the text we know that d is the smallest positive integer which is in the form asnt for st E Z Since we now have 1 in this form and 1 is the smallest positive integer then we must have d 1 In class we de ned Un to be the set of positive integers less than n which are relatively prime to n We claimed that if we consider multiplication modulo n on Un then this is a group We saw that the operation is associative and that 1 is an identity and that the Homework problem above guarantees that there are inverse elements The one question that remains which was left as an exercise for you is whether multiplication modulo 71 takes two elements in Un and gives another element in We now explain why this is true but be sure you think about this on your own before reading on Values of character sums for nite unitary groups Nathaniel Thiem and C Ryan Vinrootlf Abstract A known result for the nite general linear group GLnqu and for the nite unitary group Unqu2 posits that the sum of the irreducible character degrees is equal to the number of symmetric matrices in the group Fulman and Guralnick extended this result by considering sums of irreducible characters evaluated at an arbitrary conjugacy class of GLnqui We develop an explicit formula for the value of the permutation character of U2nqu2 over Sp2nqu evaluated an an arbitrary conjugacy class and use results con cerning GelfandGraev characters to obtain an analogous formula for Unqu2 in the case where q is an odd prime These results are also given as probabilistic statements 1 Introduction An important topic of interest in probabilistic group theory is the study ofthe statistical behavior of random conjugacy classes Fulman and Guralnick study this question for the nite general linear group by evaluating several character sums at arbitrary elements One of the main tools used there is a model for the group GLnqu which is a way of writing the sum of all of the irreducible characters of the group as a sum of characters which are induced from linear characters of subgroups The model for GLnqu is obtained by a sum of Harish Chandra induction of Gelfand Graev characters and permutation characters of the nite symplectic group in various Levi subgroups Values of the Gelfand Graev characters of the nite general linear group are known and the values of the permutation character of the nite symplectic group are obtained using results of GL2nqu conjugacy in Sp2nqu due to Wall 22 The purpose of this paper is to make the parallel computations for the nite unitary group Unqu2 which involve several structural differences from As in 8 we rely on a model for the nite unitary group and the values for the permutation character of the nite symplectic group The model for the nite unitary group involves replacing Harish Chandra induction by the more general Deligne Lusztig induction and can be found in 20 To compute the permutation character we use 22 and 7 to translate the conjugacy class information from Sp2nqu to U2nqu2 in the case of GLnqu the corresponding results are explicitly in 22 The main results obtained are only proven for odd q because the decomposition of the permutation character of U2n qu2 on Sp2nqu is not known for the case that q is even An analogous decomposition for the case q even would immediately imply the results here extend to all q The organization of the paper is as follows Section 2 reviews de nitions and results on par titions and symmetric functions with a particular emphasis on the Hall Littlewood symmetric University of Colorado at Boulder thiemn coloradoedu lUniversity of Arizona vinroot matharizonaedu MSC 2000 20033 051305 Keywords nite unitary group character sums conjugacy HallLittlewood functions functions Section 3 describes the conjugacy classes of the nite general linear unitary and symplectic groups and gives the sizes of centralizers in terms of the combinatorial information which parameterize these conjugacy classes The main results are in Section 4 In particular Theorem 43 computes the value of the permutation character U2nqu2 lndspanm 1 at an arbitrary class of U2nqu2 and Theorem 43 evaluates the sum X xelrrUnqu 2 evaluated at an arbitrary class of Un quz Finally we translate these main results into prob abilistic statements in Section 5 Acknowledgements Both authors thank the referee for helpful comments 2 Partitions and symmetric functions This section will review some fundamental de nitions and results used in this paper including partitions Hall polynomials and symmetric functions 21 Partitions Let 73 U 7 where 73 partitions of n20 For 1 11 12 11 E 73 the length V of 1 is the number of parts 1 of 1 and the size M of 1 is the sum of the parts n Let 1 denote the conjugate of the partition 1 We also write 1 1m1ltvgt2m2ltvgt where My j e 221 ll m Let 0zz denote the number of odd parts of 1 and de ne nz to be no 2971 J The following will be used in calculating a sign in one of the main results Lemma 21 Let 1 E 73 be such that either miz E 2220 wheneveri is even or miz E 2220 wheneveri is odd Then lll 0l2 E llVl2l 7104071051 2 Proof An alternate statement of the Lemma is if 1 satis es one of the conditions above and M is even then nz oV2 is even and if 1 satis es the rst condition above and M is odd then nz 0V 7 12 is even First suppose that mi1 is even whenever i is odd so that M and 0V are both even This implies that exactly half of the odd parts of 1 are of the form 1127 Consequently 7100 012 2071 0002 J 2 is even giving the result Now suppose that mil is even whenever i is even and further suppose that M is even so that oz is even as well Then exactly half of the even parts of 1 are of the form 1127 which implies that exactly half of the odd parts of 1 are of this form as well Again we have that nz oV2 is even Finally suppose that mil is even whenever i is even but that M is odd so that oz is odd This implies that l V is also odd Now choose an odd number k such that k g 11 and de ne a new partition 17 by 17 14112 lk Now 17 satis es the condition that 77745 is even whenever i is even except now we have that 1171 is even and 017 oz 1 is even as well From the previous case we have that nz7 o172 is even We also have nl7 7 nl 6k and oz72 7 0V 7 12 1 are odd Therefore nz 0V 7 12 is even D 22 Hall polynomials Let R be a discrete valuation ring with maximal ideal 50 and with nite residue eld of size q Finite R modules are then parameterized by partitions where the module M of type E 73 is isomorphic to l EBBW i1 Given a nite R module M of type the number of submodules N of type ILL such that MN is of type 1 is a polynomial in q see 17 Chapter 11 and is the called the Hall polynomial written 91q We may thus consider the Hall polynomial in some indeterminate 25 92105 Note that we have 93m 0 unless M M w and W c In particular we have 1 if 1 7 t 21 gmlt 0 otherwise where 0 denotes the empty partition 23 The ring of symmetric functions A symmetric polynomial f E Zz1m2 zn is a polynomial which is invariant under the action of the symmetric group Sn permuting the variables Let An be the set of symmetric polynomials in 2amp1 m2 mn so that An Z12 mnSquot Following 17 12 let A denote the homogeneous elements in An of degree k and for m gt n we have the natural projection map k k k pm 3 Am A Am which may be used to form the inverse limit M himXi 3 We then de ne A the ring of symmetric functions over Z in the countably in nite list x 1 2 of independent variables to be EB k and for t 6 Ex let At Zt Z A 24 HallLittlewood symmetric functions A symmetric function over Zt of central importance here is the Hall Littlewood symmetric function PAzt where A E 73 A thorough discussion of Hall Littlewood functions may be found in 17 Chapter 111 but we give the de nition here for completeness Let A E 73 such that A n where we let A 0 ifi gt A For any a 6 Sn and polynomial f in 1 zn let 0f denote the action of a on f by permuting the variables The Hall Littlewood symmetric polynomial is de ned to be WW 1 t t i L i A A 95139 95739 PA1nt 7H Z gltx11xn 120 71 066 K Then the Hall Littlewood symmetric function Pmt is obtained by nding the image of the Hall Littlewood symmetric polynomials through the limiting process described above The set of functions PAm t as A ranges over all partitions forms a Zt basis of At 17 11127 Hall polynomials show up as coef cients when products of Hall Littlewood functions are expressed as a sum of Hall Littlewood functions as given in the following result found in 17 1113 Lemma 22 t PMmtt Pyzt Zggxlwvaowi A673 We will use the following identity which is obtained in 17 1113 Example 1 Lemma 23 A I 1 m Zt l1lt1tl ygtaltztgt 11 A673 7391 igt1 We may also view a partition A E 73 as a set of ordered pairs i j of positive integers where 1 g j Ai and 1 g i A For a parameter t we de ne the function CAt as cw H 1799139 22 iiEAi7 Mei even 7 For the pair i j E A the quantity A ij is called the arm of i j and A 7i is called the leg of ij Note that we also have cm H H 14739 23 i 739 odd J39szfA The following identity involving Hall Littlewood functions and the function CAlttgt was proven by Kawanaka in 13 and a purely combinatorial proof was later given by Fulman and Guralnick in Theorem 21 Kawanaka 17 mm tlt0Agt lAlgt20AtPAz t H A6273 is 17 mmjt mko Even fork add The next identity7 also involving Hall Littlewood functions and ct7 is due to F ulman and Guralnick 87 Theorem 287 and is crucial to obtaining our main result Theorem 22 Fulman7 Guralnick CAtP t 7 1 mit 17 mimj A t0ll2 1 xi H17 zizjt iSJ39 mk Even for k Even 3 The groups and their conjugacy classes After reviewing the de nitions of the nite classical groups and their orders7 this section analyzes how their conjugacy classes interact and computes the corresponding centralizer subgroup sizes 31 The nite classical groups Let in GLnqu be the general linear group with entries in the algebraic closure of the nite eld qu with q elements Let F in A in W H 1 be the usual F robenius homomorphism Then the nite general linear groups are given by GLn qu StabnFk The nite unitary groups are given by Unrq2 aij 6 0n A s N 1 H H A 2 S V 34 and let w wij E in be an element such that wgi 1wij 1 J7 whose existence is guaranteed by the Lang Steinberg Theorem Then the nite symplectic groups are given by SPQMFq aij E GLQWFq l aijJaji J aij E U2nqu2 l aijw71waji wilewaijw71 E GL2nqu Using the second de nition above7 we see that Sp2nqu may be viewed as a subgroup of U2nqu2 We will only deal with nite orthogonal groups see for example 9 for a de nition in the case that q is odd When m is even 0m Hg and O m M are the split and non split nite orthogonal groups respectively and when m is odd Omqu is the unique nite orthogonal group The orders of the groups of interest in this paper are lGLWFqH q 12q 1q 11q 1 lUW lF12H QWLTIWWL 1nqnil1n71 q 1 lSmeFqH 7612 1q2 1 1 C12 7 1 lOiQMFqH 2qquot21i q quotq2quot 1 1mm 7 1 C12 7 1 l02n 1M 7 MW 71gtltq2lt 1gt71gtltq2 71gt 32 Conjugacy classes This section examines the conjugacy classes of the following groups and the behavior of conju gacy classes when considering one group as it is contained in another GL2nqu2 GL2nqu U2nqu2 Sp2nqu De ne sets lt1gt Q1 where for k e 221 lt1 Fk orbits in 1 and 7 i i Q2 A Q2 Q2 7 orb1ts in Q2 where S 81 39 8k H g 8 t 39 wsgq Q a Q i i i Q 7 orbits in Q where S 81 39 3k H 8 8117 39 39 qsgl Note that there is an injective map Subsets of G1 TF X q s 8182Sk gt gt f5 X731X732X73k that sends Qk to the quk irreducible polynomials in quk Thus we can identify the sets of orbits Qk Q2 and Q with sets of polynomials in the variable X Also for s 31 3 we write 3 7 3 sg A Q partition A Q a 73 is a function which assigns a partition A to each orbit s E Q We may also think of a Q partition as a sequence of partitions indexed by Q The size of A is W 2 WWW SEQ where lsl is the size of the orbit 3 Let Pd U 733 where 73 Q partitions of size n20 We can de ne Pk partitions g partitions and ltIgt partitions similarly By Jordan rational form see 17 lV2 Conjugacy classes pk of GLn M lt P such that the conjugacy class corresponding to A E 73 has characteristic polynomial H fl 591 Furthermore Wall 22 and Ennola 5 established Conjugacy classes 2 of Unqu2 lt P such that the conjugacy class corresponding to A E 73 igt2 has characteristic polynomial H fl seltigt2 The following Lemma addresses the relationship between the conjugacy classes of these groups By a slight abuse of notation we will write 11 ltIgt and 7171 ltIgt So 8 1 corresponds to the polynomial X 7 1 and s 71 to X 1 Lemma 31 a The element g E GLnqu2 in the conjugacy class gluen by A E 7332 is conjugate to an element of GLnqu if and only if pltgusqgt A for s U sq E 1 de nes a P partltlon u b The element g E GLnqu2 in the conjugacy class gluen by A E 7332 is conjugate to an element in Unqu2 if and only if pltsuggt A for s U 5 6 i2 de nes a i partltlon u c Let q be odd The element g E GL2nqu in the conjugacy class gluen by A E 73 is conjugate to an element in Sp2nqu if and only if ultsuygt A for s U 3 6 IV de nes a ltIgt partltlon u and mjA1 mjA 1 E 2220 for all oddj E 221 d Let q be odd The element g E U 2nlF 2 in the conjugacy class given by A 6 Pi is 1 2n conjugate to an element in Sp2nqu if and only if ultsuygt A for s U 3 E If de nes a ltIgt partition u and mjA1 mjA 1 E 2220 for all oddj E 221 Proof Part a follows from considering Jordan rational forms or elementary divisors as in 17 lV2 while parts b and c are due to Wall 22 d Suppose g E U2nqu2 p is well de ned and and mjA1 mjA 1 E 2220 for all odd j E 221 Since g E GL2nqu2 b implies there exists I E 7333 such that A 117 for s r U 77 6 2 However by assumption there is a ltIgt partition 397 such that QWETLU 117 for r U W E I is well de ned Thus g is conjugate to an element in GL2nqu and the result follows from c Suppose g E U2nqu2 is conjugate to an element in Sp2nqu Since Sp2nqu Q GL2nqu we have that g is conjugate to an element of GL2nqu and the result follows from D We note that in each of these groups the corresponding conjugacy class is unipotent exactly when AU is the only nonempty partition of the Pk partition or g partition For the rest of this section we let q be odd The conjugacy classes of Sp2nqu are not parameterized by ltIgt partitions A of size 2n such that mjA1 mjA 1 E 2220 for all odd j E 221 as might be suggested by Lemma 31 Instead we need to distinguish between some classes in Sp2n R that may be conjugate in GL2n M The development we give here follows W A symplectic signed partition is a partition with a function 6 2 2i 6 2221 l m2i 7g 0 i1 such that mj E 2220 for all oddj E 221 For example v 554141333322211gt 5244342t3712 is a symplectic signed partition where the parts of size 4 are assigned the sign 7 and the parts of size 2 the sign For even i the multiplicity ofi in 39y mi39y will be given the sign that the partsi are assigned In the example above we have m439y 72 sgnm4 7 m2 Y 2 and sgnm239y Let Pi denote the set of all symplectic signed partitions If 39y is a symplectic signed partition we let 39yquot denote the partition obtained by ignoring the signs of the sets of even parts of 39y We de ne various functions on symplectic signed partitions as its value on 39yquot for example we de ne M and n39y as l39yol and n39y respectively De ne 73 as the set of symplectic signed partitions of size n Let PM denote the set of functions A ltIgt a 73 U Pi such that a A1A 1 6 Pi are symplectic signed partitions b AW 6 7D for r e qr 7 7e i1 For A E Pig de ne W Z W WT L 79 and let P A 6 P l w 271 Then from results of Wall 22 Conjugacy classes My of Sp2nqu lt P2 such that the conjugacy class corresponding to A E Piqy has characteristic polynomial H fr 79 Alternatively we may consider the set PM of functions 397 I a 73 U Pi such that a duo 1 6 Pi are symplectic signed partitions 10 77 70 for all r E I c 77 6 73 for r E I r 7 i1 If we de ne 7 and 732i analogously then the set 732i also parameterizes the conjugacy classes of Sp2n M In this case if g is an element of Sp2nqu in the conjugacy class corresponding to 397 E Pig then the conjugacy class ofg in GL2nqu is given by A E 73 where A 70 for r i1 and A 77 otherwise 33 Centralizers For g in a group G let 009 h E G l 9h hgh denote the centralizer of g in G Theorem 31 Ennola Wall Let 9 E Unqu2 be in the conjugacy class indexed by u E 7332 The order of the centralizer of g in Unqu2 is a 1l l H aye quL where M96 WWW H 190 99in i j1 for u 177L12m23m3 6 73 Let q be odd and let 397 E 73ft For each r E I de ne Ar y7i as follows where we let UmiFqlTl if r 7 r 7 i1 lGLmivqulrll12 if T 3 V T 7 i17 3391 lsp in i1 2 odd 14 7 NJ 7 mm qlmilZlosgnltmigtlmlrq if r i1 2 even 9 In the case that r i1 and i and mi are both even sgnmi says whether to take the split or non split orthogonal group and when m is odd this will always give the unique orthogonal group regardless of the sign of mi For each r E we de ne Br39yT as 31 70gt qlrll7 l2n7m72 m32gt H Ar77i 32 These factors are de ned in order to state the following result of Wall 22 which tells us the size of a given conjugacy class of Sp2n qu Theorem 32 Wall Let g E Sp2nqu be in the conjugacy class parameterized by 7 E 7324 Then the order of the centralizer ofg in Sp2nqu is given by lep2nrq9l H 30377 r6 Remark We have applied the following identity for partitions in order to state the results of Wall in the form given in Theorem 32 For any 39y E 73 we have 1 ZWMWJ39W 5 Eh 1miv2 Wl2 71W ZWWWQ iltj i 139 Note that an element 5 6 2 satis es either 5 6 2 or 5 U U 17 where 1217 6 2 with v 7 17 For I E 73 and 5 6 2 i1 let UmiFq2lsl 12 if 5 is odd 1 2 i i AS 119 lGLmquisil 1f 5 is even 5 7 5 UmiFqlsl if5UU17 UU GLmiFqlsl2 if5UU17 U17 B5I9 qlsllv l2nvs72inf2 H A5u9 33 where mi nub9 Theorem 33 Let g E Sp2nqu be in the U2nqu2 conjugacy class parameterized by I 6 733 and in the Sp2nqu conjugacy class parameterz39zed by 7 E 732 Then the order of the centralizer ofg in Sp2nqu is given by lCSp2nqugl B1771B717771 H B57VS39 562i1 Proof From Theorem 32 we just need to prove that H Br77 H B5I5 34 Tamil 56ltigt2i1 Suppose 5 6 2 i1 satis es 5 6 2 Note that since 5 is odd the relation 5 5 would imply the existence of some 5139 E 5 such that 571 51 However i1 5 so such an 5139 cannot exist Thus 5 7 5 Let r 5U 5 so that r E r r and lrl 2l5l Then 77 119 HM and by de nition EmaW 33 V9B5 um 10 Suppose that s 17 U 17 with 17 31 17 E 12 and that s 31 3 Let r 17 U 17 so that 1 17 U 17 31 1 Then we have T E I M lsl 119 77 and Emalt1 Bltsultsgtgt and Bltmltgtgt W W Finally suppose that s 17 U 17 with 17 31 17 E 12 and s 3 Either 17 17 or 17 17 If 17 17 then in fact we have s so that s E I Letting r s we have 77 119 and BMW Bltsultsgtgt lf17 17 then we have 17 E I and 17 31 17 Letting r 17 we have M lsl2 and 119 77 70 so that 3 lt1 vlt1gtgtBltrnltTgtgt Be W This exhausts all factors in both sides of 34 and the desired result is obtained El 4 Character sums This section contains our main results including formulas for Uaquz IndSp2nqu 1g for g E U2nqu2 and Xg for g E Unqu2 XelrrUnqu2 in Theorem 43 and Theorem 44 respectively 41 Deligne Lusztig induction and a model for Unqu2 For any class function X of Unqu2 and any conjugacy class of Unqu2 corresponding to I E 7332 we let XI denote the value of X evaluated on any element of the conjugacy class corresponding to V Let Pm denote the Gelfand Graev character of Unqu2 which by 21 takes values given by the character formula 7 il n24 1 7 iqy if p is unipotent and u 111 POM T 0 otherwise 41 C a On where On class functions of Un quz n20 be the ring of class functions with multiplication given by Deligne Lusztig induction Um n X07 RUmUX E 42 where X E Cm and E E On See 2 for example for a discussion of Deligne Lusztig induction This ring structure is studied in 4 and it turns out to be exactly the same structure induced by a product on class functions of Un using Hall polynomials which was studied by Ennola In particular given pI E 73 we de ne the Hall polynomial 92105 by s 921475 H 92ltsgtyltsgttlsl 43 56ltigt2 The following result is proven in 20 but also seems to be implicit in 11 Theorem 41 LetX E Cm 7 E Cm and let 6 Then the value of X05 on the conjugacy class A of Um nqu2 is given by X 0 EW Z 9y qXl 539 HEPSZJIEPS2 In the case that q is odd Henderson 10 has decomposed the permutation character lndgzzn into irreducible constituents and proved that it is multiplicity free It is well known that the Gelfand Graev character is multiplicity free see 19 for example and the decomposition into irreducibles for the nite unitary group is given in terms of Deligne Lusztig characters in 3 and in terms of irreducibles in 18 Using these results along with the characteristic map of the nite unitary group the following result is proven in 20 Theorem 42 Let q be odd and let Un Unqu2 192k Sp2kqu and let 1 be the trivial character The sum of all distinct irreducible complex characters of Un is given by X 2 PM o Indg1 XeiuU k21n Giving the sum of all distinct irreducible characters of a group as a sum of onedimensional characters induced from subgroups is called a model for the group In Theorem 42 Deligne Lusztig induction is used in place of subgroup induction so this is a slight variant from the classical situation Note that since Deligne Lusztig induction does not in general produce characters it is somewhat surprising that all of these products do in fact result in characters A model for the group GLnqu was rst found by Klyachko 15 and made more explicit by lnglis and Saxl 12 and the result for that case is analogous to Theorem 42 except with Deligne Lusztig induction replaced by parabolic induction 42 Values of the permutation character Let Un Unqu2 and 5192a Sp2nqu The goal of this section is to use results from Section 3 to give a closed formula for the value of 1 Indgggnu evaluated at an arbitrary conjugacy class of U2 Throughout this section we let q be the power of an odd prime Let g 6 U2 From the formula for an induced character we have a 6 U2 agail E Spg C n i n sz g29 h E 51927 h g 1n U2n 44 TL 71 U n 151272ng 7 If g is in the conjugacy class indexed by I E 73 then the value of CUM av as given by Theorem 31 So we need to know the number of elements h in Spgn which are conjugate to g in U2 Let s 6 i1 and let I E 73 Writing m raj119 de ne As V9i and Bs 119 as follows lmiZl Amman H lt17 1M qrz mZDHWZJ Sammm j1 Bsy9 1Wl2nltvltsgtgtoltvltsgtgt2HMSyum 45 12 Theorem 43 Let I E 7353 and let q be odd Then 111 0 unless i For every 3 6 i2 119 Vs ii For every odd j Z 1 mjI1mjI 1 E 2220 If both and ii are satis ed then all H9915 BSv 1157 where a1 is as de ned in Theorem 31 and Bs7 119 is de ned as in and Proof Lemma 31 d and 44 imply that 1gnI 0 if I does not satisfy and ii else the conjugacy class parameterized by I is not conjugate to an element in Spgn Let h E Spgn7 and suppose h is in the Ugn conjugacy class parameterized by I E 733 Thus7 by Lemma 31 d7 if h is in the Spgn conjugacy class parameterized by u E 732 then U n 1322 V pmo 111 Mime 114 and Msuy 115 for all s U 5 E If 11 From 44 and Theorem 337 we have all szn l5an 317 MB17 Y H96ltTgt2i1 387 119 U2 1517 V de i Honchoqeo a1 1 mews Bltwltsgtgt 317MB17V39 So we want to show 1 Z 46 1 317Vlt1gtB717Vlt 1gt Blt1MgtBlt71vgt39 HWEPi toylt1gtyoylt71gt Note that we can think of the set of symplectic signed partitions Pi as Pi 176 E 73 x S7 where 5 6 Z 6 2221 I 2 for some 7 1 so that if a 111 and 39y 114 then equation 46 becomes 4 7 Z 317MB17 Y 565 317M76B17r39 T657 From 31 and 327 we have for s E 11 and 76 6 Pi 31370175 q 2nM Zi quotLE21114137 7 5M qWWWEZH ismqu H qmiZlo wmtm iodd ieven where mi mia For t odd7 miZ miZ lSpmi7qul 61 4 11qu 1 qmimmgZ H 1 1612 j1 j1 13 Note that H qmiZ q02 iodd From 46 and 45 it suffices to show n D even WOW2 21 even WWW2 q H 47 Hi even A17M71A717 V71 1417 M7 671A717 3977 T7 6651 139 even T657 Let s 6 i1 and 116 6 Pi For i even and m mi1b odd 2 Uni21 I 2 Uni21 I Newmww wm mmmmIh enem evwgt j1 j1 For i even and m even 1487 whiz qmi2105imnFq1 Wig271 ewwwwhwgt ltwen j1 Wig271 2qltmi221 M WZ H 1 1M2 j1 Returning to 47 let M even 1 miu 73 01 even 1 mi39y 37 Then 1 565 even 1417 M7 67 Z14ak717 77 7 7 57 i 2i even WWWPD even WWVZgiM q 1miM1 WW1 1 1 1 1 Hi even M7 014717 V7 2 illn q q e e n mm even mini even 1 1 HEEWWE g a gwmm ea lt56 T65 mm even mm even In the last sum there is a choice of signs for 6 and 739 Whenever i is even and the multiplicities are nonzero and so there are 2M terms in the sum De ne Me even 1 mi1b 37 0 and mi1b is even1 1 even 1 mi39y 37 0 and mi39y is even1 M0 even 1 mi1b is odd1 1 even 1 mi1b is odd1 so that M Me 1 M0 Since the choices of sign for m odd do not affect the sum in 48 we have 1 H 1 see MHzWW 1nqemm2 7396 5w man even mm even 1 1 2M0 49 n 11 1 SiWW 1 Tjquw i7miHw7wmj7 even WW even mjm even where there are exactly 2ME terms in the second sum For any set of numbers 0417 7ak an induction argument gives us the identity k 1 2k k 2 39 51525ke41j11 670quot Hj11 0 Applying this identity to 497 we obtain 1 2M0 1 Tjqmi39l2 1 m39 2 H 2 J 41 i even 1 llq 200 jeven hmwmm w even WW eVen ev M r 2 r 410 Himin even1 1q WW 11mm even1 1qlmz7l Substituting 410 into 487 we obtain the desired result 47 CI Theorem 43 has an especially nice form when evaluated at unipotent elements Corollary 41 Let uy 6 Un be a unipotent element of type a Unless 777401 E QZgt0 whenever 4 U2 7 7 2 2s odd 15p2nu 7 0 and otherwzse we have U2 WW CI nll0v2 lsPZWWD 7 Band 7 q CMlt 1q39 Corollary 41 immediately follows from Theorem 43 and the following result Lemma 41 Let E 73 Then UH w nltAgtltweoltAgtgt2 71 71 BM lt gt q cm qgt Proof Recall the de nition of et as given in 22 From the form of this function given in 237 we have i 39 H 1135 gtlt17 flqr i 39 39 H my M lt1 e flc027 The de nitions of a7q in Theorem 31 and of B17 in 45 now imply the result 1 milq 43 Values of the full character sum We now calculate the sum of all characters of Un at an arbitrary element Theorem 44 Let q be odd and let A E 7332 Then ZXSIHWR x 0 unless i For every 3 6 i2 M9 A9 ii For every odd l mi 1 E 2220 iii For every even i mi1 E 2220 If i ii and iii are satis ed then 0A1 lt gt a Z X XEIITUT HSE R BS7 AltSgtgt where a is as de ned in Theorem 31 and Bs9 is de ned as in and 15 Proof Let A E 73g From Theorems 41 and 42 we have 2 w 2 womdzzgmw XEIrrUn 217 1 Z gzyequmlgzgxw k21n H6352 167332 From Equation 41 we have F000 0 unless u is unipotent so we may assume 1 M1 where M k and M9 0 when 3 7 1 As in 43 we have S QSVPQ H 92ltsgtyltsgtqlsl 56ltigt2 Since M9 0 when 3 7 1 then from 21 we have that if s 7 1 9 Ms A 7 5 1 1fI g sl s q 0 otherwise From these facts and from the value of PM given in 41 we have XW XEIITUT k21n nepk 0 lt1 1 i 2 92V q1W21 Ml iq 70111 ygpi2ylt1gt1p 1 VSASs1 Since M 1 Am and 1 is even in the above sum we have 11ll1l21 1911421 11lvl2 and 11l1l 11M and so we multiply the outside of the sum by the expressions on the left sides of the equations above and the inside of the sum by the expressions on the right sides We have 119 A when 3 7 1 and from Theorem 43 1211 0 unless M5 AM for s E 12 mi 1 is even when i is odd and miI1 is even when i is odd and otherwise we have i Hse g B57 115 Hs1BSAS B1 17 22101 7 Hs1 1WSl 5 aw HAM M761 where 1 111 So ZXSIHWW x 0 unless M5 AM for s E I and mi 1 is even when i is odd and otherwise we have lt gt lt gt s s s X11llll21lx1lH11l H laAS1ql l XEIITUn 911 4 MZ W 1 1 l l 1 1 i M Egyq1Mlllq s va M i1 V H5744 B57 mku even forkodd From Lemma 22 with t 71q this expression is the coef cient of PA1m71q in the expansion of n ltgt ltgt ltgt s 5 5 Lg 11llll21lx1 1H1l H laASql l 91 Z Ptmil qlelw uemy Z P lt 1qgt 1gt lt qgt n i1 1e H m lt7qgtnltvgt Bum 1111 Be W fgr k odd By applying Lemma 23 with t 71q and y q and replacing each M with in this expression is the coef cient of PA1 71q in the expansion of s 1 i qWWU1ll1l21l1lH71l llslaks7qlsl H 91 13921 T 9 Emilq 1 2ayq 761 B17 1 H5744 387 Am 673 mm even for k odd Applying Lemmas 41 and 21 this expression simpli es to n H H H sS 9 179 Lg A11llx1 l21l11H1lx H laxwaiq l H 1 m 91 13921 1 qlvliovZCV1q Pl zgi1 lt M HelBlteAltsgtgt p mw even for k odd We may now apply Theorem 21 with t 71q to change the last sum into a product which simpli es the above expression to n H H H s 5 5 q A11ll1lZll11H1l H laksql l 91 H 1 7 mg 1 7 mimj 1 121 1 x 1914 ziqu Hf1 Bss Finally Theorem 22 with t 71q implies that the coef cient of PA1z 71q of the above expression is 0 unless ka m is even whenever k is even in which case it is 1 1 no WIN2mm WM 191 191 3Alt1gt1 1gt 1gt0A MA W2 g 71gt HH aw H s 91 H9744 B37 A Applying Lemmas 41 and 21 and simplifying gives the desired result 1 In the case that we are nding the sum of the characters of Un at a unipotent element we obtain a result much like Corollary 41 Corollary 42 Let My 6 Un be a unipotent element of type In Unless 777401 E 2220 whenever i is even ZXSIHWR x01 0 and otherwise we have 1 Ma 7 004 WW W qnlttgtltMolttgtgt2CM1q xelrrUn 7 u Proof This follows immediately from Theorem 44 and Lemma 41 D 17 5 Probabilities and Frobenius Schur indicators Here we reinterpret the main results in Sections 42 and 43 as probabilistic statements as the corresponding results for GLn M are given in For the result in Theorem 43 we have the following statement Corollary 51 Let q be odd The probability that a uniformly randomly selected element of Sp2nqu belongs to the conjugacy class in U2nqu2 corresponding to I E 733 is 0 unless 9 I y for euery s E 12 and raj111 and mill71 are euen for euery oddj Z 1 in which case it is equal to 1 Hse g BSv 115 Proof From Equation 44 we have for any 9 6 U2 0 U 7 Un g A 15127249 7 Hh E Spgn h N g in U273 Taking g to be in the conjugacy class corresponding to I 6 733 we have CU2ngl al from Theorem 31 So the probability we want is exactly 517 U i 1 2 V 7 HhESPZn l hwgln all 51an l The result now follows directly from Theorem 43 D Interpreting the result for the full character sum in Theorem 44 as a probabilistic statement is most easily accomplished using the twisted F robenius Schur indicator originally de ned in 14 and further studied in Let G be a nite group with automorphism L which either has order 2 or is the identity Let 7139 V be an irreducible complex representation of G and suppose that L7139 E 7139 where 7139 is the contragredient representation of 7139 and L7139 is de ned by L7139g 7139Lg The equivalence of the representations L7139 and 7139 implies that there exists a nondegenerate bilinear form ltVXVgtC unique up to scalar by Schur s Lemma such that lt7r9v 7r9wgt ltv7wgt7 for every 9 E G uw E V Because the bilinear form is unique up to scalar we have ltv7wgt 6i7rltw7 vgt7 where 647139 i1 depends only on L and 7139 If L7139 7139 we de ne 647139 0 Then 647139 is the twisted Frobenius Schur indicator of 7139 V with respect to L If X is the character corresponding to 7139 we also write 647139 64X The following results are proven in 14 and 1 and show that the twisted F robenius Schur indicator indeed generalizes the classical F robenius Schur indicator which is the case that L is trivial Theorem 51 Let G be a nite group with automorphism L such that L2 is the identity Then we have the formulas 1 SAX 2X0 by and Z 6igtltgtlt9 H71 E G l h 71 gll 960 XEIITltGgt Now consider the case G Unqu2 and L the transpose inverse automorphism In 20 it is proven that SAW 1 for every irreducible complex representation 7139 V of Unqu2 By Theorem 51 this is equivalent to 20 Corollary 52 x1 Hg 6 Unqu2 l g symmetricl 51 X611TUnqu2 We conclude with the following probabilistic version of Theorem 44 Corollary 52 Let g be odd Let u be a uniformly randomly selected element of Unqu2 and let L be the transpose inuerse automorphism of Unqu2 The probability that ubu is in the conjugacy class corresponding to I E 7332 is 0 unless 119 y for every 3 6 2 raj111 is euen for euery euen j and mill71 is euen for euery odd j in which case it is equal to q01 Hseltigt2 B57 s39 Proof Let g E Ugn be an element from the conjugacy class corresponding to V From Theorem 51 and the fact that SAW 1 for every irreducible representation 7139 of U we have 2 xI 6 Un l uLu gl XEIITUT We need to count each set on the right as g ranges over the conjugacy class corresponding to V which has size lUgnlal by Theorem 31 Multiplying by this quantity and dividing by lenl to get a probability we nd that the desired probability is Mu 6 Un l Wu gll Emma X01 all all The result now follows from Theorem 44 CI References 1 D Bump and D Ginzburg Generalized F robenius Schur numbers J Algebra 278 2004 no 1 2947313 2 R Carter Finite groups of Lie type conjugacy classes and complex characters John Wiley and Sons 1985 3 P Deligne and G Lusztig Representations of reductive groups over nite elds Ann of Math 2 103 1976 no 1 1037161 4 F Digne and J Michel Foncteurs de Lusztig et caracteres des groupes lineaires et unitaires sur un corps ni J Algebra 107 1987 no 1 2177255 19

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