Cohen-Lenstra heuristics and random matrix theory over finite fields

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1 Cohen-Lenstra heurstcs and random matrx theory over fnte felds Jason Fulman Department of Mathematcs, Unversty of Southern Calforna, Los Angeles, CA, E-mal address: Abstract. Let g be a random element of a fnte classcal group G, and let λ z (g) denote the partton correspondng to the polynomal z n the ratonal canoncal form of g. As the rank of G tends to nfnty, λ z (g) tends to a partton dstrbuted accordng to a Cohen-Lenstra type measure on parttons. We gve sharp upper and lower bounds on the total varaton dstance between the random partton λ z (g) and the Cohen-Lenstra type measure.. Introducton The study of conjugacy classes of random elements of a group s an actve subject. Indeed, for symmetrc groups two elements are conjugate f and only f they have the same cycle structure, so ths amounts to the study of the cycle structure of random permutatons. And for the untary groups U(n, C), two elements are conjugate f and only f they have the same set of egenvalues, so ths amounts to the study of egenvalues of random matrces. Motvated by these consderatons (and unaware of the Cohen-Lenstra heurstcs of number theory), the author, n a seres of papers [F], [F3], [FG], [FST], nvestgated the conjugacy classes of random elements of a fnte classcal group. Two elements of GL(n, q) are conjugate f and only f they have the same ratonal canoncal form [H]. Moreover, f g s an element of a fnte classcal group, there s a partton λ z (g) correspondng to the polynomal z n the ratonal canoncal form of g. Lettng the rank of G tend to nfnty, we proved that ths random partton has a lmtng dstrbuton. For example f G = GL(n, q), one obtans the dstrbuton P GL on the set of all parttons of all non-negatve ntegers, whch chooses λ wth Key words and phrases. random matrx, random partton, Cohen-Lenstra heurstc. 00 AMS Subject Classfcaton: 5B5, 60B0. Date: June 9, 03. Fulman was partally supported by NSA grant H

2 JASON FULMAN probablty P GL (λ) = ( /q ) Aut(λ), where Aut(λ) denotes the automorphsm group of a fnte abelan group of type λ. We became fascnated wth the combnatorcs of such random parttons arsng from random matrx theory over fnte felds. In the papers [F], [F3], [FG], [FST], we lnked them to the Hall-Lttlewood polynomals of symmetrc functon theory, and developed and appled probablstc algorthms for growng such random parttons. We were delghted to recently learn from Lengler [L] that our work on random parttons s related to the Cohen-Lenstra heurstcs [CL] of number theory. Indeed, Cohen and Lenstra study random parttons chosen wth probablty ( /q ) Aut(λ), exactly the same formula as n our constructon n the GL case. It s beyond the scope of ths paper to offer any sort of survey of Cohen-Lenstra heurstcs, but we can assure the reader that research n the area s actve and ongong, wth contrbutons from Bhargava, Malle, Ellenberg, Venkatesh, Poonen, Rans, and many others. We can recommend the papers [D], [EV], and the many references theren. We are confdent that all of the random parttons studed n the current paper wll turn out to be related to Cohen-Lenstra heurstcs. Indeed, our random parttons n the symplectc case were recently redscovered n the Cohen-Lenstra context [Ac]. One of the goals of the current paper s to collect n one place all of the formulas for random parttons arsng from random matrces over fnte felds. These are currently scattered n the lterature. A second goal of the current paper s to quantfy the convergence of the random parttons λ z (g) to ther lmt dstrbutons. Recall that the total varaton dstance P Q T V between two probablty dstrbutons on a set X s defned as P Q T V = P (x Q(x). x X Let Λ GL,z,n denote the measure on parttons of sze at most n arsng by takng the partton correspondng to the polynomal z n the ratonal canoncal form of a random element of GL(n, q). One of the results of ths paper s the sharp bound:.38 q n+ P GL Λ GL,z,n T V q n+. Ths s more explct and sharper than a smlar recent result of Maples [Map], though we note that Maples man nterest s dfferent than ours: he proves unversalty of the dstrbuton P GL for matrx ensembles where the

3 Cohen-Lenstra heurstcs and random matrx theory 3 entres are d, but not necessarly unform. We prove smlar sharp bounds for the fnte untary, symplectc, and orthogonal groups, n both odd and even characterstc (where thngs can dffer). In terms of future work, t would be worthwhle to further explore the connectons n [F] made between symmetrc functon theory (Hall-Lttlewood polynomals) and random parttons arsng from fnte classcal groups. In fact n work complementary to ours, Okounkov [O], [O], [O3] makes many nterestng connectons between symmetrc functon theory (but not Hall- Lttlewood polynomals) and random parttons (but not Cohen-Lenstra type measures). In fact one of hs constructons, namely the defnton of random parttons from Macdonald polynomals, was made ndependently n [F]. It would be very nterestng to adapt Okounkov s methods to our settng. The organzaton of ths paper s as follows. Secton treats random parttons arsng from the fnte general lnear groups. The untary case s treated n Secton 3 and the symplectc case s treated n Secton. Secton 5 treats random parttons arsng from the fnte orthogonal groups, and s splt nto two subsectons, whch consder odd and even characterstc respectvely.. General lnear groups The Cohen-Lenstra measure [CL] s a probablty dstrbuton on the set of all parttons of all non-negatve ntegers. We denote ths measure by P GL, snce analogs for other fnte classcal groups wll be gven n other sectons. A formula for the measure P GL s: P GL (λ) = ( /q ) Aut(λ). Here Aut(λ) denotes the automorphsm group of a fnte abelan group of type λ. Page 8 of [Mac] gves the followng explct formula: Aut(λ) = q (λ ) (/q) m (λ). Here m (λ) s the number of parts of λ of sze, and λ s the partton dual to λ n the sense that λ = m (λ) + m + (λ) +. Also (/q) j denotes ( /q)( /q ) ( /q j ). Remark: The measure P GL s the specal case (u = ) of a measure studed n [F], whch chooses λ wth probablty ( u/q u λ ) Aut(λ).

4 JASON FULMAN Ths probablty can be rewrtten as ( u/q ) Pλ(u/q, u/q, u/q 3, ; /q) q n(λ), where P λ denotes a Hall-Lttlewood polynomal and n(λ) = ( λ ). Ths suggests the study of more general measures based on Macdonald polynomals; see [F] for detals, and for probablstc algorthms for generatng parttons accordng to ths measure. Next recall the ratonal canoncal form of an element g GL(n, q). Ths s dscussed at length n Chapter 6 of the textbook [H], and corresponds to the followng combnatoral data. To each monc non-constant rreducble polynomal φ over F q, one assocates a partton (perhaps the trval partton) λ φ of some non-negatve nteger λ φ. Let deg(φ) denote the degree of φ. The only restrctons necessary for ths data to arse from an element of GL(n, q) are that λ z = 0 and φ λ φ deg(φ) = n. We are nterested n the partton (of sze at most n) correspondng to the polynomal z n the ratonal canoncal form of a random element of GL(n, q), and we let Λ GL,z,n denote the correspondng measure on parttons. From [F0], t s known that as n, the measure Λ GL,z,n converges to the measure P GL. The man result of ths secton gves sharp bounds for ths convergence. The next lemma s due to Euler; see page 9 of [An]. Lemma.. () ( u/q ) = j 0 u j q (j ) (q j )(q ). () ( u/q = j 0 ( u) j (q j )(q ). A proof of the followng lemma s contaned n [St]. Lemma.. λ u λ Aut(λ) = ( u/q. Next we obtan an explct expresson for P GL Λ GL,z,n T V. Proposton.3. P GL Λ GL,z,n T V = q (m ) (q m ( /q ) ) (q ) + n q (m ) (q m ) (q ) ( /q ( ) j (q j ) (q ).

5 Cohen-Lenstra heurstcs and random matrx theory 5 Proof : Frst we consder the contrbuton to the total varaton dstance comng from λ of sze m > n. Snce Λ GL,z,n (λ) = 0 for such λ, the contrbuton s ( /q ). Aut(λ) λ = By Lemma. and part of Lemma., ths s equal to q (m ) (q m ) (q ) ( /q ). Next consder the probablty that Λ GL,z,n assocates to λ when λ = m n. By the cycle ndex of the general lnear groups [F], ths probablty s equal to the coeffcent of u n Aut(λ) φ z,z [ µ u deg(φ) µ Aut(µ) q q deg(φ) Multplyng and dvdng by u µ µ Aut(µ) gves that ths s equal to the coeffcent of u n [ ] u deg(φ) µ = Aut(λ) Aut(λ) µ µ u µ Aut(µ) u µ Aut(µ) φ z u. µ ]. Aut(µ) q q deg(φ) The equalty followed snce settng all varables equal to one n the cycle ndex of GL(n, q) yelds /( u). From Lemma., ths s the coeffcent of u n Thus Λ GL,z,n (λ) s equal to Aut(λ) ( u/q ). u Coef. u j n Aut(λ) whch by part of Lemma. s equal to ( u/q ), ( ) j Aut(λ) (q j ) (q ). It follows that the contrbuton to P GL Λ GL,z,n T V comng from λ wth λ = m n s n Aut(λ) ( /q ( ) j (q j ) (q ). λ =m.

6 6 JASON FULMAN By Lemma. and part of Lemma., λ =m Aut(λ) = Coef. um n ( u/q = q (m ) (q m ) (q ). Thus the contrbuton to P GL Λ GL,z,n T V from λ wth λ = m n s n q (m ) (q m ) (q ) ( /q ( ) j (q j ) (q ). Ths completes the proof. Next we prove the man result of ths secton. Theorem.. For n,.38 q n+ P GL Λ GL,z,n T V q n+. Proof : To begn we consder the lower bound. By consderng the m = n+ term n Proposton.3, t follows that P GL Λ GL,z,n T V q (n+ ) (q n+ ( /q ) ) (q ) = q n+ ( /qn+ )( /q n+3 ). Snce n and q, ths s at least.38/q n+. Note that we have used the bound ( / j ) = 8 ( / j ) 3 j 3 j 8 3 ( / / + / 5 + / 7 / / 5 ).77, whch follows from Euler s pentagonal number theorem, stated on page of [An]. Next we consder the upper bound. Frst note that q (m ) (q m ( /q ) ) (q ) = = q (m ) q (m+ ) q m q n+ ( /q) q n+.

7 Cohen-Lenstra heurstcs and random matrx theory 7 Second, note that by applyng part of Lemma. wth u =, n q (m ) (q m ) (q ) ( /q ( ) j (q j ) (q ) = n q (m ) (q m ) (q ) (q + ) (q ) n q (m ) q (m+ ) ( /q) ( /q m ) q (+ ) ( /q) ( /q + ) Snce ( /q 3.5, ths s at most (3.5) n q m+(+ ) (3.5) q n+ ( /q) 3 q n+. The upper bound of the theorem follows mmedately by combnng the bounds n the prevous paragraph. 3. Untary groups Next we defne a untary analog P U of the Cohen-Lenstra measure on the set of all parttons of all non-negatve ntegers. A formula for the measure P U s: P U (λ) = + /( q) ( ) Aut U (λ). Here Aut U (λ) s defned by the formula Aut U (λ) = q (λ ). ( /q) m (λ), where ( /q) j = ( + /q)( /q ) ( ( ) j /q j ). Remark: The measure P U s the specal case (u = ) of a measure studed n [F], whch chooses λ wth probablty ( + u/( q) u λ ) Aut U (λ). To defne the probablty measure Λ U,z,n, we use, as n the GL case, the theory of ratonal canoncal forms. Gven an element g U(n, q), there s a partton λ z (g) of sze at most n assocated to the polynomal z. When g s chosen unformly at random from U(n, q), we let Λ U,z,n denote the

8 8 JASON FULMAN correspondng measure on parttons. From [F0], t s known that as n, the measure Λ U,z,n converges to the measure P U. The man result of ths secton makes ths quanttatve, provng that 6q n+ P U Λ U,z,n T V 3 q n+. Lemma 3. s obtaned by replacng u, q by u, q n Lemma.. Lemma 3.. () ( + u/( q) ) = ( ) (j+ ) u j j 0 (q j ( ) j )(q+). () ( + u/( q) = u j q (j ) j 0 (q j ( ) j )(q+). The next lemma s a untary analog of Lemma.. Lemma 3.. λ u λ Aut U (λ) = ( + u/( q). Proof : From the formulas for Aut(λ) and Aut U (λ), one checks that u λ Aut U (λ) = ( u) λ. Aut(λ) q q λ λ The result now follows from Lemma., snce settng u u, q q n ( u/q yelds ( + u/( q). Next we gve an explct expresson for P U Λ U,z,n T V. Proposton 3.3. = P U Λ U,z,n T V q (m ) (q m ( ) m ) (q + ) + n ( + /( q) q (m ) (q m ( ) m ) (q + ) ( + /( q) ) ( ) (j+ ) (q j ( ) j ) (q + ) Proof : Frst we consder the contrbuton to the total varaton dstance comng from λ of sze m > n. Snce Λ U,z,n (λ) = 0 for such λ, the contrbuton s ( + /( q) ). Aut U (λ) λ = By Lemma 3. and part of Lemma 3., ths s equal to q (m ) (q m ( ) m ) (q + ) ( + /( q) )..

9 Cohen-Lenstra heurstcs and random matrx theory 9 Next consder the probablty that Λ U,z,n assocates to λ when λ = m n. By the cycle ndex of the untary groups [F], ths probablty s equal to the coeffcent of u n Indeed, Aut U (λ) ( u λ ( u λ u λ Aut U (λ) u λ Aut U (λ) s the part of the cycle ndex of the untary groups correspondng to polynomals other than z. By Lemma 3., t follows that Λ U,z,n (λ) s equal to the coeffcent of u n Aut U (λ) and hence equal to ( + u/( q) ), u Coef. u j n Aut U (λ) By part of Lemma 3., ths s equal to. ( + u/( q) ). ( ) (j+ ) Aut U (λ) (q j ( ) j ) (q + ). Thus the contrbuton to P U Λ U,z,n T V comng from λ wth λ = m n s n Aut U (λ) ( + /( q) ( ) (j+ ) (q j ( ) j ) (q + ). λ =m λ =m By Lemma 3. and part of Lemma 3., = Coef. u m n ( + u/( q) Aut U (λ) = q (m ) (q m ( ) m ) (q + ). Thus the contrbuton to P U Λ U,z,n T V from λ wth λ = m n s n q (m ) (q m ( ) m ) (q + ) ( + /( q) ( ) (j+ ) (q j ( ) j ) (q + ), and the proof s complete.

10 0 JASON FULMAN Next we prove the man result of ths secton. Theorem 3.. For n, 6q n+ P U Λ U,z,n T V 3 q n+. Proof : To start we examne the lower bound. By consderng the m = n + term n Proposton 3.3, t follows that P U Λ U,z,n T V q (n+ ) ( + /( q) ) q (n+ ) ( + /q)( /q ) ( ( ) n+ /q n+ ) q (n+ ) ( /q) q (n+ ) ( + /q) ( /q) = ( + /q) q n+ 6q n+. Next we treat the upper bound. Frst note that = q n+. q (m ) (q m ( ) m ( + /( q) ) ) (q + ) q (m ) ( + /( q) ) q (m+ ) ( + /q) ( ( ) m /q m ) q (m ) q (m+ ) Next note by part of Lemma 3. that ( + /( q) ( ) (j+ ) (q j ( ) j ) (q + ) = ( ) (j+ ) (q j ( ) j ) (q + ) j + q (j+ ) j + q (+ ).

11 Thus Cohen-Lenstra heurstcs and random matrx theory n ( + /( q) n q (m ) (q m ( ) m ) (q + ) q (m+ n q (m ) ( ) (j+ ) (q j ( ) j ) (q + ) ) ( + /q) ( ( ) m /q m ) q m q (+ ) q (+ ) q n+ ( /q) q n+. Combnng the bounds n the two prevous paragraphs completes the proof.. Symplectc groups Next we defne a symplectc analog P Sp of the Cohen-Lenstra measure on the set of all parttons of all non-negatve ntegers n whch the odd parts occur wth even multplcty. A formula for the measure P Sp s: P Sp (λ) = /q ( ) Aut Sp (λ), where Aut Sp (λ) s defned by the formula λ n(λ)+ Aut Sp (λ) = q + o(λ) ( /q )( /q ) ( /q m (λ) ). Here, m (λ) denotes the multplcty of n the partton λ, o(λ) denotes the number of odd parts of λ, and n(λ) = ( λ ). Remark: The measure P Sp s the specal case (u = ) of a measure studed n [F3], whch chooses λ wth probablty ( u /q u λ ) Aut Sp (λ). To defne the probablty measure Λ Sp,z,n, we use, as n the GL and U cases, the theory of ratonal canoncal forms. Gven an element g

12 JASON FULMAN Sp(n, q), there s a partton λ z (g) of sze at most n assocated to the polynomal z. When g s chosen unformly at random from Sp(n, q), we let Λ Sp,z,n denote the correspondng measure on parttons. From [F3] (n odd characterstc) and [FG] (n even characterstc), t s known that as n, the measure Λ Sp,z,n converges to the measure P Sp. In fact the man result of ths secton s that. q n+ P Sp Λ Sp,z,n T V.5 q n+. Lemma. s obtaned by replacng u by u q and q by q n Lemma.. Lemma.. () ( u /q ) = j 0 () ( u /q = j 0 The followng lemma s from [F3]. Lemma.. λ u λ Aut Sp (λ) = u j q j (q j )(q ). ( u /q. ( ) j u j q j (q j )(q ). Now we gve an explct expresson for P Sp Λ Sp,z,n T V. Proposton.3. P Sp Λ Sp,z,n T V = q m (q m ) (q ( /q ) ) + n q m (q m ) (q ) ( /q ( ) j q j (q j ) (q ). Proof : To begn we consder the contrbuton to the total varaton dstance comng from λ of sze m > n. Snce Λ Sp,z,n (λ) = 0 for such λ, the contrbuton s ( /q ). Aut Sp (λ) λ = By Lemma. and part of Lemma., ths s equal to q m (q m ) (q ) ( /q ). Next consder the probablty that Λ Sp,z,n assgns to λ when λ = m n. By the cycle ndex of the symplectc groups, ths probablty s equal to

13 Cohen-Lenstra heurstcs and random matrx theory 3 the coeffcent of u n Aut Sp (λ) Indeed, ( u λ ( u u λ λ Aut Sp (λ) u λ Aut Sp (λ) s the part of the cycle ndex of the symplectc groups correspondng to polynomals other than z. By Lemma., t follows that Λ Sp,z,n (λ) s equal to the coeffcent of u n and thus equal to Aut Sp (λ) ( u/q ), u Coef. u j n Aut Sp (λ) By part of Lemma., ths s equal to. ( u/q ). ( ) j q j Aut Sp (λ) (q j ) (q ). Thus the contrbuton to P Sp Λ Sp,z,n T V comng from λ wth λ = m n s n Aut Sp (λ) ( /q ( ) j q j (q j ) (q ). λ =m λ =m By Lemma. and part of Lemma., = Coef. u m n ( u /q Aut Sp (λ) = q m (q m ) (q ). Thus the contrbuton to P Sp Λ Sp,z,n T V from λ wth λ = m n s n q m (q m ) (q ) ( /q ( ) j q j (q j ) (q ), whch completes the proof. Now we prove the man result of ths secton.

14 JASON FULMAN Theorem.. For n,. q n+ P Sp Λ Sp,z,n T V.5 q n+. Proof : To begn we treat the lower bound. By lookng at the m = n + term n Proposton.3, t follows that P Sp Λ Sp,z,n T V q (n+) q (n+ ) For the upper bound, frst note that = q n+.. q n+. ( /q ) q m (q m ) (q ( /q ) ) ( /q ) q m ( /q ) ( /q m ) q m Next, note that by part of Lemma. wth u =, Snce n q m (q m ) (q ) ( /q ( ) j q j (q j ) (q ) n q m q + (q m ) (q ) (q (+ ) (q ). the upper bound becomes (.5) n ( /q )( /q ).5, q m (.5) q (+) q n+ ( /q ).5 q n+. Combnng the bounds of the prevous two paragraphs completes the proof.

15 5. Orthogonal groups Cohen-Lenstra heurstcs and random matrx theory 5 In treatng the orthogonal groups t s necessary to separately consder the cases of odd and even characterstc. Subsecton 5. treats odd characterstc, and Subsecton 5. treats even characterstc. 5.. Odd characterstc. Suppose throughout ths subsecton that the characterstc s odd. We defne an orthogonal analog P O of the Cohen-Lenstra measure on the set of all parttons of all non-negatve ntegers n whch the even parts occur wth even multplcty. A formula for the measure P O s: P O (λ) = where Aut O (λ) s defned by the formula λ n(λ)+ Aut O (λ) = q o(λ) ( /q ), Aut O (λ) ( /q )( /q ) ( /q m (λ) ). Here, as earler, m (λ) denotes the multplcty of n the partton λ, o(λ) denotes the number of odd parts of λ, and n(λ) = ( λ ). Remark: The measure P O s the specal case (u = ) of a measure studed n [F3], whch chooses λ wth probablty ( u /q ) u λ ( + u) Aut O (λ). To defne the probablty measure Λ O,z,n, we use, as n the cases of other fnte classcal groups, the theory of ratonal canoncal forms. We choose an element g, wth probablty / unformly at random from O + (n, q) and wth probablty / unformly at random from O (n, q). Then there s a partton λ z (g) of sze at most n assocated to the polynomal z. We let Λ O,z,n denote the correspondng measure on parttons. From [F3] t s known that as n, the measure Λ O,z,n converges to the measure P O. The man result of ths secton s a sharp error term for ths convergence. The followng lemma s from [F3]. Lemma 5.. Suppose that q s odd. Then λ u λ Aut O (λ) = + u ( u /q ). Next we gven an explct expresson for P O Λ O,z,n T V.

16 6 JASON FULMAN Proposton 5.. Suppose that q s odd. Then P O Λ O,z,n T V = + + n q m / (q m ) (q ) q (m ) / (q m ) (q ) q m / (q m ) (q ) ( /q + n ()/ q (m ) / (q m ) (q ) ( /q ()/ ( /q ) ( /q ) ( ) j q j (q j ) (q ) ( ) j q j (q j ) (q ). Proof : To begn we consder the contrbuton to the total varaton dstance comng from λ of sze m > n. Snce Λ O,z,n (λ) = 0 for such λ, the contrbuton s λ = Aut O (λ) ( /q ). By Lemma 5. and part of Lemma., ths s equal to + q m / (q m ) (q ) q (m ) / (q m ) (q ) ( /q ) ( /q ). Next we consder the probablty that Λ O,z,n assgns to λ when λ = m n. By the cycle ndex of the orthogonal groups [F], ths s equal to Aut O (λ) Coef. u n Indeed, the term +u cycle ndex, and the term ( u /q ) ( u /q ) + u ( u /q ) ( u /q ) u. corresponds to the polynomal z + n the u corresponds to polynomals other

17 Cohen-Lenstra heurstcs and random matrx theory 7 than z ± n the cycle ndex. Cancelng terms, one obtans that Λ O,z,n (λ) s equal to Aut O (λ) Coef. u n ( u /q ) u = Aut O (λ) = Aut O (λ) = Aut O (λ) ()/ ()/ Coef. u j n Coef. u j n ( u /q ) ( ) j q j (q j ) (q ), ( u /q ) where the last step used part of Lemma.. Thus the contrbuton to P O Λ O,z,n T V comng from λ wth λ = m n s n λ =m Aut O (λ) ( /q By Lemma 5. and part of Lemma., f m s even, and λ =m λ =m ()/ Aut O (λ) = q m/ (q m ) (q ), Aut O (λ) = q (m )/ (q m ) (q ), f m s odd. Thus the contrbuton to P O Λ O,z,n T V wth λ = m n s n q m / (q m ) (q ) ( /q + n ()/ q (m ) / (q m ) (q ) ( /q ()/ ( ) j q j (q j ) (q ). ( ) j q j (q j ) (q ) ( ) j q j (q j ) (q ). comng from λ

18 8 JASON FULMAN Ths completes the proof. Now we prove the man result of ths secton. Theorem 5.3. () For n even and q odd, () For n odd and q odd,. q n/ P O Λ O,z,n T V.3 q n/.. q (n+)/ P O Λ O,z,n T V q (n+)/. Proof : Suppose that n s even. To lower bound the total varaton dstance, lookng at the m = n + term n Proposton 5. gves that q n / P O Λ O,z,n T V (q n ) (q ). q n/. q n / q n /+n/ ( /q ) ( /q ) Next we consder the upper bound when n s even; by Proposton 5. ths s a sum of four terms. The frst term s = q m / (q m ) (q ) q m/ q m/ = q n/+ ( /q) 3 8 q n/+ 8q n/. ( /q ) ( /q ) ( /q ) ( /q m )

19 Cohen-Lenstra heurstcs and random matrx theory 9 The second term n the upper bound s = q (m ) / (q m ) (q ) q (m )/ q (m )/ = q n/ ( /q) 3 8q n/. ( /q ) ( /q ) ( /q ) ( /q m ) The thrd term n the upper bound s n q m / (q m ) (q ) ( /q n q m / ()/ ( ) j q j (q j ) (q ) q (+)/ (q m ) (q ) (q + ) (q ). Snce j ( /qj., the thrd term s at most (.) n q m/ q ( + ) (.) q n/+ ( /q) 3(.) 8q n/+. q n/.

20 0 JASON FULMAN The fourth term n the upper bound s n q (m ) / (q m ) (q ) ( )/ ( /q ( ) j q j (q j ) (q ) n q (m ) / q (+)/ (q m ) (q ) (q + ) (q ). Agan usng that j ( /qj., the fourth term s at most (.) n q (m )/ q (.) ( + ) q n/ ( /q).6 q n/. Combnng the above bounds proves the theorem for n even. Next suppose that n s odd. To lower bound the total varaton dstance, lookng at the m = n + term n Proposton 5. gves that P O Λ O,z,n T V q (n+) / (q n+ ) (q ( /q ) ) q (n+)/ ( /q ). q (n+)/. By Proposton 5., the upper bound s a sum of four terms. For the frst term, one argues as n the n even case to obtan q m / (q m ) (q ) ( /q ) q m/ = q (n+)/ ( /q) 3 8q (n+)/.

21 Cohen-Lenstra heurstcs and random matrx theory For the second term, one also argues as n the n even case to obtan q (m ) / (q m ) (q ( /q ) ) q (m )/ The thrd term n the upper bound s n q m / (q m ) (q ) ( /q n q m / ( )/ = q (n+)/ ( /q) 3 8q (n+)/. ( ) j q j (q j ) (q ) q (+)/ (q m ) (q ) (q + ) (q ). Usng j ( /qj. shows that the thrd term s at most (.) n. q m/ q ( + ) q (n+)/ ( /q) The fourth term n the upper bound s n q (m ) / (q m ) (q ).6 q (n+)/. ()/ ( /q ( ) j q j (q j ) (q ) n q (m ) / q (+)/ (q m ) (q ) (q + ) (q ). Usng j ( /qj. shows that the fourth term s at most (.) n q (m )/ (.) q ( + ) q (n+)/ ( /q).6 q (n+)/.

22 JASON FULMAN Combnng the above bounds proves the theorem for n odd. 5.. Even characterstc. Throughout ths subsecton t s assumed that the characterstc s even. We defne an orthogonal analog P O of the Cohen- Lenstra measure on the set of all parttons of all non-negatve ntegers n whch the odd parts occur wth even multplcty. A formula for the measure P O s: P O (λ) = where Aut O (λ) s defned by the formula λ n(λ)+ Aut O (λ) = q + o(λ) l(λ) ( /q ), Aut O (λ) ( /q )( /q ) ( /q m (λ) ). Here, as earler, m (λ) denotes the multplcty of n the partton λ, o(λ) denotes the number of odd parts of λ, and n(λ) = ( λ ). The symbol l(λ) denotes the number of parts of λ. Remark: The measure P O s the specal case (u = ) of a measure studed n [FST] whch chooses λ wth probablty ( u /q ) u λ + u Aut O (λ). To defne the probablty measure Λ O,z,n, we use, as n the other cases, the theory of ratonal canoncal forms. We choose an element g, wth probablty / unformly at random from O + (n, q) and wth probablty / unformly at random from O (n, q). (Note that n even characterstc, odd dmensonal orthogonal groups are somorphc to symplectc groups, so we focus on even dmensonal orthogonal groups). Then there s a partton λ z (g) of sze at most n assocated to the polynomal z. We let Λ O,z,n denote the correspondng measure on parttons. From [FST] t s known that as n, the measure Λ O,z,n converges to the measure P O. The man result of ths secton s to make ths convergence quanttatve. The followng lemma s from [FST]. Lemma 5.. Suppose that q s even. Then λ u λ Aut O (λ) = + u ( u /q ). Next we gve an explct expresson for P O Λ O,z,n T V.

23 Cohen-Lenstra heurstcs and random matrx theory 3 Proposton 5.5. Suppose that q s even. Then P O Λ O,z,n T V [ ] = q m (q m ) (q ) + q (m ) (q m ) (q ) ( /q ) [ + n m= ( /q + ( /q q m (q m ) (q ) + q (m ) (q m ) (q ) ( ) j q j (q j ) (q ) n ( ) j q j (q j ) (q ). Proof : To begn, we consder the contrbuton to the total varaton dstance comng from λ of sze m > n. Snce Λ O,z,n (λ) = 0 for such λ, the contrbuton s ( /q ). Aut O (λ) λ = By Lemma 5. and part of Lemma., ths s equal to [ ] q m (q m ) (q ) + q (m ) (q m ) (q ( /q ). ) Next consder the probablty that Λ O,z,n assocates to λ when λ = m n. By the cycle ndex of the orthogonal groups [FST], ths probablty s equal to the coeffcent of u n Indeed, Aut O (λ) ( u /q ) u. ( u /q ) u s the part of the cycle ndex of the orthogonal groups correspondng to polynomals other than z. Thus Λ O,z,n (λ) s equal to Coef. u j n Aut O (λ) ( u/q ). ]

24 JASON FULMAN By part of Lemma., ths s equal to ( ) j q j Aut O (λ) (q j ) (q ). Thus the contrbuton to P O Λ O,z,n T V comng from λ wth λ = m n s n Aut O (λ) ( /q ( ) j q j (q j ) (q ). λ =m By Lemma 5. and part of Lemma., f m, then λ =m Aut O (λ) = Coef. u m n + u ( u /q ) = q m (q m ) (q ) q (m ) + (q m ) (q ). Thus the contrbuton to P O Λ O,z,n T V comng from λ wth λ = m wth m n s [ ] n q m (q m ) (q ) + q (m ) (q m ) (q ) m= ( /q ( ) j q j (q j ) (q ). The contrbuton to P O Λ O,z,n T V comng from λ = 0 s n ( /q ( ) j q j (q j ) (q ), whch completes the proof. Next we prove the man result of ths secton. Theorem 5.6. For n and q even,. q n P O Λ O,z,n T V.6 q n

25 Cohen-Lenstra heurstcs and random matrx theory 5 Proof : To lower bound the total varaton dstance, lookng at the m = n+ term n Proposton 5.5 gves that q n P O Λ O,z,n T V (q n ) (q ) q n q n(n+) ( /q ). q n. ( /q ) Next we consder the upper bound; by Proposton 5.5, ths s a sum of three terms. Snce q m (q m ) (q ) q (m ) (q m ) (q ), the frst term s at most q (m ) (q m ) (q ( /q ) ) = ( /q ) q m ( /q ) ( /q m ) q m = q n ( /q) q n. To upper bound the second term, note that [ ] n q m (q m ) (q ) + q (m ) (q m ) (q ) m= ( /q ( ) j q j (q j ) (q ) m q (m ) (q m ) (q ) m= ( /q ( ) j q j (q j ) (q ) n q (m ) q + (q m ) (q ) (q (+ ) (q ). m=

26 6 JASON FULMAN Snce j ( /qj.5, the second term s at most (.5) n m= q m (.5) q (+) q n ( /q ).5 q n. To upper bound the thrd term, note that n ( /q ( ) j q j (q j ) (q ) q n+ (q (n+ ) (q ) Snce j ( /qj.5, the thrd term s at most.5.5 q (n+) 3 q n.05 q n. Addng the upper bounds on the three terms completes the proof. References [Ac] Achter, J., The dstrbuton of class groups of functon felds, J. Pure Appl. Algebra 0 (006), [An] Andrews, G., The theory of parttons, Addson-Wesley, Readng, Mass., 976. [CL] Cohen, H. and Lenstra, H.W., Jr., Heurstcs on class groups of number felds, n Number theory, Noordwjerhout 983, 33-6, Lecture Notes n Math. 068, Sprnger, Berln, 98. [D] Delaunay, C., Heurstcs on class groups and on Tate-Shafarevch groups: the magc of the Cohen-Lenstra heurstcs, n Ranks of ellptc curves and random matrx theory, 33-30, London Math. Soc. Lecture Notes Ser. 3, Cambrdge Unv. Press, Cambrdge, 007. [EV] Ellenberg, J., and Venkatesh, A., Statstcs of number felds and functon felds, n Proceedngs of the Internatonal Congress of Mathematcans. Volume II, 383-0, Hndustan Book Agency, New Delh, 00. [F0] Fulman, J., Probablty n the classcal groups over fnte felds: symmetrc functons, stochastc algorthms and cycle ndces, Ph.D. thess, Harvard Unversty, 997. [F] Fulman, J., Cycle ndces for the fnte classcal groups, J. Group Theory (999), [F] [F3] [FG] Fulman, J., A probablstc approach toward conjugacy classes n the fnte general lnear and untary groups, J. Algebra (999), Fulman, J., A probablstc approach to conjugacy classes n the fnte symplectc and orthogonal groups, J. Algebra 3 (000), 07-. Fulman, J. and Guralnck, R., Conjugacy class propertes of the extenson of GL(n, q) generated by the nverse transpose nvoluton, J. Algebra 75 (00), [FST] Fulman, J., Saxl, J., and Tep, P. H., Cycle ndces for fnte orthogonal groups of even characterstc, Trans. Amer. Math. Soc. 36 (0), [H] Hersten, I.N., Topcs n algebra, Second edton. Xerox College Publshng, Lexngton, Mass.-Toronto, Ont., 975. [L] Lengler, J., The Cohen-Lenstra heurstc: methodology and results, J. Algebra 33 [Mac] (00), Macdonald, I., Symmetrc functons and Hall polynomals, Second edton. The Clarendon Press, New York, 995.

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