## Random matrix theory over finite fields

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Venue: | Bull. Amer. Math. Soc. (N.S |

Citations: | 22 - 6 self |

### BibTeX

@ARTICLE{Fulman_randommatrix,

author = {Jason Fulman},

title = {Random matrix theory over finite fields},

journal = {Bull. Amer. Math. Soc. (N.S},

year = {},

pages = {51--85}

}

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### Abstract

Abstract. The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with symmetric function theory, Markov chains, Rogers-Ramanujan type identities, potential theory, and various measures on partitions.

### Citations

1985 | Random Graphs
- Bollobas
- 1985
(Show Context)
Citation Context ... given in Subsection 3.1. Analogous universality results are known for matrices with complex entries [So]. For further information on the rank of random 0 − 1 matrices, see [BKW] for sparse matrices, =-=[Bo]-=- for a survey of results on the rank over the real numbers, and also the discussion of work of Rudvalis and Shinoda in Subsection 3.2. Elkies [El] studies the rank of Hankel random matrices with non-u... |

1560 |
Exactly Solved Models in Statistical Mechanics
- Baxter
- 1982
(Show Context)
Citation Context ...but one still wants a motivation for trying to write the left hand side of the Andrews-Gordon identity in product form. The best motivation is Baxter’s work on statistical mechanics (surveyed in [A2],=-=[Bax1]-=-,[Bax2]) in which he really needed “sum = product” identities and was led to conjecture analogs of Rogers-Ramanujan type identities. Although a proof of the Rogers-Ramanujan identities doesn’t emerge ... |

455 |
Symmetric functions and Hall polynomials, second edition
- Macdonald
- 1995
(Show Context)
Citation Context ...obability of belonging to the corresponding conjugacy class and is therefore equal to one over the order of the centralizer of a representative. It is well known (e.g. easily deduced from page 181 of =-=[Mac]-=-) that one over the order of the centralizer of the conjugacy class of GL(n, q) corresponding to the data {λφ} is 1 � φ qdeg(φ)·P i (λ′ φ,i )2 � i≥1 ( 1 q deg(φ) ) mi(λφ) The formulas given for conjug... |

368 | On the distribution of the length of the longest increasing subsequence of random permutations - Baik, Deift, et al. - 1999 |

345 |
The art of computer programming. Vol. 2: Seminumerical algorithms
- Knuth
- 1998
(Show Context)
Citation Context ...Running times of algorithms). One of the main approaches to computing determinants and permanents of integer matrices involves doing the computations for reductions mod prime powers. Section 4.6.4 of =-=[Kn]-=- gives a detailed discussion with references to literature on upper bounds of running times. If one believes that typical matrices one encounters in the real world are like random matrices, this motiv... |

318 |
and condition numbers of random matrices
- Edelman, “Eigenvalues
- 1989
(Show Context)
Citation Context ... fields. In fact von Neumann’s interest in eigenvalues of random matrices with independent normal entries arose from the same heuristic applied to questions in numerical analysis (the introduction of =-=[Ed]-=- gives further discussion of this point). Examples of algorithms in which properties of random matrices were really needed to bound running times include recognizing when a group generated by a set of... |

153 | Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem
- Aldous, Diaconis
- 1999
(Show Context)
Citation Context ...st increasing subsequences in non-uniform random permutations [F3]. It is beyond the scope of this paper to survey the literature on longest increasing subsequences, but an accessible introduction is =-=[AlDia]-=-. In what follows Ja(q) is the polynomial discussed on pages 52-54 of [F1], h(s) denotes the hook-length of a box in λ [Mac] and [n] = qn−1 q−1 is the q-analog of the number n. Recall that a skew diag... |

150 | Discrete orthogonal polynomial ensembles and the Plancherel measure
- Johansson
(Show Context)
Citation Context ... be made precise and led to a solution of the long-standing conjecture relating lengths of increasing subsequences of permutations to eigenvalues of random matrices. For these developments see [BOOl],=-=[Jo]-=- and the many references therein. Now we return to the measure MGL,u,q and describe an algorithm for growing random partitions according to this measure. The Young Tableau Algorithm Step 0: Start with... |

145 | Asymptotics of Plancherel measures for symmetric groups
- Borodin, Okounkov, et al.
(Show Context)
Citation Context ...tic can be made precise and led to a solution of the long-standing conjecture relating lengths of increasing subsequences of permutations to eigenvalues of random matrices. For these developments see =-=[BOOl]-=-,[Jo] and the many references therein. Now we return to the measure MGL,u,q and describe an algorithm for growing random partitions according to this measure. The Young Tableau Algorithm Step 0: Start... |

132 |
The theory of partitions, Encyclopedia of Mathematics and its
- Andrews
- 1976
(Show Context)
Citation Context ...1ifφ = z − 1andset xφ,λ = 0 otherwise. One concludes that � 1 q λ⊢n P i (λ′ i )2 � 1 i ( q ) = mi(λ) qn(n−1) |GL(n, q)| . Now multiply both sides by un ,suminn, and apply Euler’s identity (page 19 of =-=[A1]-=-): ∞� unq (n2) (qn − 1) ···(q − 1) = ∞� 1 ( 1 − u qr ). n=0 The measure MGL,u,q is a fundamental object for understanding the probability theory of conjugacy classes of GL(n, q). This emerges from The... |

116 |
On the maximal subgroups of the finite classical groups
- Aschbacher
- 1984
(Show Context)
Citation Context ...NDOM MATRIX THEORY OVER FINITE FIELDS 61 [F1]. Shalev [Sh1] uses facts about the distribution of the order of a random matrix together with Aschbacher’s study of maximal subgroups of classical groups =-=[As]-=- as key tools in studying the probability that a random element of GL(n, q) belongs to an irreducible subgroup of GL(n, q) thatdoesnotcontainSL(n, q). As explained in [Sh1] this has a number of applic... |

112 | Universality at the edge of the spectrum in Wigner random matrices
- Soshnikov
- 1999
(Show Context)
Citation Context ...a type of “universality” result for the asymptotic description of random elements of GL(n, q) to be given in Subsection 3.1. Analogous universality results are known for matrices with complex entries =-=[So]-=-. For further information on the rank of random 0 − 1 matrices, see [BKW] for sparse matrices, [Bo] for a survey of results on the rank over the real numbers, and also the discussion of work of Rudval... |

110 |
memoir on the expansion of certain infinite products
- Rogers, Second
(Show Context)
Citation Context ...scent of transfer matrices in statistical mechanics.) Details are in [F6]. The Markov chain approach is also related to quivers [F7]. 3.5. Rogers-Ramanujan Identities. The Rogers-Ramanujan identities =-=[Ro]-=- 1+ 1+ ∞� n=1 ∞� n=1 q n2 (1 − q)(1 − q 2 ) ···(1 − q n ) = q n(n+1) (1 − q)(1 − q 2 ) ···(1 − q n ) = ∞� n=1 ∞� n=1 1 (1 − q 5n−1 )(1 − q 5n−4 ) 1 (1 − q 5n−2 )(1 − q 5n−3 ) are among the most intere... |

104 |
q-Series: their development and applications in analysis, number theory, combinatorics, physics, and computer algebra, CBMS series 66
- Andrews
- 1986
(Show Context)
Citation Context ...u,q is a probability measure. Theorem 11 diagonalizes the transition matrix K, finding a basis of eigenvectors, which is fundamental for understanding the Markov chain (part 3 is stated as a lemma in =-=[A2]-=-). Since the matrix K is upper triangular with distinct eigenvalues, this is straightforward. Theorem 11. 1. Let C be the diagonal matrix with (i, i) entry ( 1 u q )i( q )i. LetM � � be the matrix .Th... |

96 | Random matrix theory and ζ(1/2 - Keating, Snaith |

94 |
A view of random number generators
- Marsaglia
- 1985
(Show Context)
Citation Context ... of finite simple quotients of PSL(2,Z); a group G is a quotient of PSL(2,Z)ifand only if G =< x,y>with x2 = y3 = 1. For further discussion, see [Sh2]. Example 7 (Random number generators). We follow =-=[Mar]-=-,[MarTs] in indicating the relevance of random matrix theory to the study of random number generators. Suppose one wants to test a mechanism for generating a random integer between 0and233 − 1. In bas... |

86 |
Regular elements of semisimple algebraic groups
- Steinberg
- 1965
(Show Context)
Citation Context ...to that of Theorem 1 shows that the n →∞probability that an element of GL(n, q) is cyclic is (1− 1 q 5 )/(1+ 1 q 3 ). For large q this goes like 1−1/q 3 . The reason for this is a result of Steinberg =-=[Stei]-=- stating that the set of non-regular elements in an algebraic group has co-dimension 3 (see also [GuLub]). In type A, regular (i.e. centralizer of minimum dimension) and cyclic elements coincide, but ... |

74 | Generating random elements of a finite group
- Celler, Leedham-Green, et al.
- 1995
(Show Context)
Citation Context ...tional. Given a generating set S of a finite group G, it is natural to seek random elements of G. One approach, implemented in the computer systems GAP and MAGMA, is the product replacement algorithm =-=[CeLgMuNiOb]-=-. Fixing G and some k, one performs a random walk on k-tuples (g1, ··· ,gk) ofelementsofGwhich generate the group. The walk proceeds by picking an ordered pair (i, j) with1≤i�= j ≤ n uniformly at rand... |

67 |
Multiple series Rogers-Ramanujan type identities
- Andrews
- 1984
(Show Context)
Citation Context ...abilistic proof of the Rogers-Ramanujan identities. To illustrate the idea we give the proof of the following generalization of the first Rogers-Ramanujan identity (called the Andrews-Gordon identity =-=[A3]-=-,[Gor]): � n1,··· ,nk−1≥0 r=1 1 q N 2 1 +···+N 2 k−1(1/q)n1 ···(1/q)nk−1 = ∞� r=1 r�=0,±k(mod 2k+1) 1 1 − (1/q) r where Ni = ni+···+nk−1. Setting k = 2 and replacing q by its reciprocal specialize to ... |

67 | The sampling theory of selectively neutral alleles. Theoretical biology 3 - Ewens - 1972 |

61 |
Identities of the Rogers-Ramanujan type
- Bailey
- 1949
(Show Context)
Citation Context ... needed both RogersRamanujan identities. Andrews’ paper [A4] notes that many proofs of the Rogers-Ramanujan identities make use of the following mysterious result called Bailey’s Lemma, alluded to in =-=[Bai]-=- and stated explicitly in [A3]. A pair of sequences {αL} and {βL} is called a Bailey pair if L� αr βL = . (1/q)L−r(u/q)L+r Bailey’s Lemma states that if α ′ L r=0 = uL q L2 αL and β ′ L = � L r=0 u r ... |

59 |
Mirror symmetry and elliptic curves,” in The Moduli Space of Curves (Texel Island
- Dijkgraaf
- 1994
(Show Context)
Citation Context ...s like, and conditioning Pq to live on partitions of size n gives a uniform partition. The measure Pq is related to the vertex operators [O1] and to the enumeration of ramified coverings of the torus =-=[Dij]-=-. In this regard the papers [O1] and [BlO] prove that the k point correlation function F (t1, ··· ,tk) = � q |λ| n� ∞� 1 λi−i+ 2 t λ k k=1 i=1 q j2 . .s76 JASON FULMAN is a sum of determinants involvi... |

59 |
Ordered cycle lengths in a random permutation
- Shepp, Lloyd
- 1966
(Show Context)
Citation Context ...er of fixed points, number of cycles, the order of a permutation, length of the longest cycle) by generating functions. We refer the reader to [Ko] for results in this direction using analysis and to =-=[ShLl]-=- for results about cycle structure proved by a probabilistic interpretation of the cycle index generating function. Historically important papers in random permutation theory are [ErT], [Gon], and [Ve... |

51 | The boundary of Young graph with Jack edge multiplicities
- Kerov, Okounkov, et al.
- 1998
(Show Context)
Citation Context ...e branchings of Definition 1 are multiplicative. Kerov [Ke2] has a conjectural description of the boundary. It has been verified for Schur functions [T], Kingman branching [Kin], and Jack polynomials =-=[KeOOl]-=-, but remains open for the general case of Macdonald polynomials. In particular, it is open for Hall-Littlewood polynomials, the case related to T (n, q). It is interesting that the κ(λ, Λ) of Definit... |

49 | On some problems of a statistical group-theory
- Erdős, Turán
- 1967
(Show Context)
Citation Context ...nalysis and to [ShLl] for results about cycle structure proved by a probabilistic interpretation of the cycle index generating function. Historically important papers in random permutation theory are =-=[ErT]-=-, [Gon], and [VeSc]. Subsection 2.1 reviews the conjugacy classes of GL(n, q) and then discusses cycle indices for GL(n, q) andMat(n, q), the set of all n × n matrices with entries in the field of q e... |

46 |
A combinatorial generalization of the Rogers-Ramanujan identities
- Gordon
(Show Context)
Citation Context ...stic proof of the Rogers-Ramanujan identities. To illustrate the idea we give the proof of the following generalization of the first Rogers-Ramanujan identity (called the Andrews-Gordon identity [A3],=-=[Gor]-=-): � n1,··· ,nk−1≥0 r=1 1 q N 2 1 +···+N 2 k−1(1/q)n1 ···(1/q)nk−1 = ∞� r=1 r�=0,±k(mod 2k+1) 1 1 − (1/q) r where Ni = ni+···+nk−1. Setting k = 2 and replacing q by its reciprocal specialize to the fi... |

46 |
Combinatorial Enumeration of Groups, Graphs and Chemical Compounds
- Pólya, Read
- 1987
(Show Context)
Citation Context ...i π∈G i≥1 and is called a cycle index because it stores information about the cycle structure of elements of G. Applications of the cycle index to graph theory and chemical compounds are exposited in =-=[PoRe]-=-. It is standard to refer to the generating function 1+ � n≥1 u n n! � � x π∈Sn i≥1 ni(π) i as the cycle index or cycle index generating function of the symmetric groups. n! From the fact that there a... |

44 |
On the conjugacy classes in the unitary, symplectic and orthogonal groups
- Wall
- 1963
(Show Context)
Citation Context ...IX THEORY OVER FINITE FIELDS 63 q 2 ). One such form is given by <�x, �y >= � n i=1 q xiyi . Any two non-degenerate skew-linear forms are equivalent, so that U(n, q) is unique up to isomorphism. Wall =-=[W1]-=- parametrized the conjugacy classes of the finite unitary groups and computed their sizes. To describe his result, an involution on polynomials with non-zero constant term is needed. Given a polynomia... |

43 |
Random partitions in population genetics
- Kingman
- 1978
(Show Context)
Citation Context ...are homeomorphic and that the branchings of Definition 1 are multiplicative. Kerov [Ke2] has a conjectural description of the boundary. It has been verified for Schur functions [T], Kingman branching =-=[Kin]-=-, and Jack polynomials [KeOOl], but remains open for the general case of Macdonald polynomials. In particular, it is open for Hall-Littlewood polynomials, the case related to T (n, q). It is interesti... |

43 | A recognition algorithm for special linear groups
- Neumann, Praeger
- 1992
(Show Context)
Citation Context ...pan V . As is explained in [NP2], this is equivalent to the condition that the characteristic and minimal polynomials of α are equal. The need to estimate the proportion of cyclic matrices arose from =-=[NP1]-=- in connection with analyzing the running time of an algorithm for deciding whether or not the group generated by a given set of matrices in GL(n, q) contains the special linear group SL(n, q). Cyclic... |

40 |
The structure of random partitions of large integers
- Fristedt
- 1993
(Show Context)
Citation Context ...em 11. ∞� urn (1 − u/q2n )(−1) n−j ( u q )n+j−1 qrn2 ( 1 u (n−j q )L−n( q )L+nq 2 ) 1 ( q )n−j Remarks. 1. One of our motivations for seeking a Markov chain description of MGL,u,q is work of Fristedt =-=[Fris]-=-, who had a Markov chain approach for the measure Pq on the set of all partitions of all natural numbers defined by Pq(λ) = � ∞ i=1 (1 − qi )q |λ| where q<1. Fristedt’s interest was in studying what a... |

40 | The product replacement algorithm and Kazhdan’s property (T
- Lubotzky, Pak
(Show Context)
Citation Context ...random matrices is crucial to their analysis. A recent effort to understand the performance of the product replacement algorithm uses Kazhdan’s property T from the representation theory of Lie groups =-=[LubPa]-=-; the paper [Pa] is a useful survey. Much remains to be done. Example 9 (Running times of algorithms). One of the main approaches to computing determinants and permanents of integer matrices involves ... |

37 |
RandomMatrix Theory and ζ(1/2+it
- Keating, Snaith
- 2000
(Show Context)
Citation Context ...x numbers, two matrices are in the same conjugacy class if and only if they have the same set of eigenvalues. Hence, at least in this case, which is related to the zeroes of the Riemann zeta function =-=[KeaSn]-=-, the study of eigenvalues is the same as the study of conjugacy classes. As complements to this article, the reader may enjoy the surveys [Py1],[Py2], [Py3],[Sh2],[Sh3] on enumerative and probabilist... |

34 | What do we know about the product replacement algorithm
- Pak
- 1999
(Show Context)
Citation Context ...crucial to their analysis. A recent effort to understand the performance of the product replacement algorithm uses Kazhdan’s property T from the representation theory of Lie groups [LubPa]; the paper =-=[Pa]-=- is a useful survey. Much remains to be done. Example 9 (Running times of algorithms). One of the main approaches to computing determinants and permanents of integer matrices involves doing the comput... |

33 |
Random Mappings. Optimization Software
- Kolchin
- 1986
(Show Context)
Citation Context ...njugacy class functions of random permutations (e.g. number of fixed points, number of cycles, the order of a permutation, length of the longest cycle) by generating functions. We refer the reader to =-=[Ko]-=- for results in this direction using analysis and to [ShLl] for results about cycle structure proved by a probabilistic interpretation of the cycle index generating function. Historically important pa... |

33 |
Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen symmetrischen Gruppe
- Thoma
- 1964
(Show Context)
Citation Context ...f these two branchings are homeomorphic and that the branchings of Definition 1 are multiplicative. Kerov [Ke2] has a conjectural description of the boundary. It has been verified for Schur functions =-=[T]-=-, Kingman branching [Kin], and Jack polynomials [KeOOl], but remains open for the general case of Macdonald polynomials. In particular, it is open for Hall-Littlewood polynomials, the case related to ... |

33 | A Rogers-Ramanujan bijection - Garsia, Milne - 1981 |

32 | A recognition algorithm for classical groups over finite fields
- Niemeyer, Praeger
- 1998
(Show Context)
Citation Context ...ment of GL(n, q) is called a primitive prime divisor (ppd) element if its order is divisible by a primitive prime divisor of q e − 1with n/2 <e≤ n. This is a conjugacy class function. The analysis in =-=[NiP]-=- derives elegant bounds on the proportions of ppd elements in the finite classical groups and applies them to the group recognition problem for classical groups over finite fields (determining when a ... |

31 |
Matrices and the structure of random number sequences, Linear Algebra and its
- Marsaglia, Tsay
- 1985
(Show Context)
Citation Context ...nite simple quotients of PSL(2,Z); a group G is a quotient of PSL(2,Z)ifand only if G =< x,y>with x2 = y3 = 1. For further discussion, see [Sh2]. Example 7 (Random number generators). We follow [Mar],=-=[MarTs]-=- in indicating the relevance of random matrix theory to the study of random number generators. Suppose one wants to test a mechanism for generating a random integer between 0and233 − 1. In base 2 thes... |

29 | Random words, Toeplitz determinants and integrable systems
- Its, Tracy, et al.
- 2001
(Show Context)
Citation Context ... which is important in representation theory and random matrix theory. Letting x1 = ··· = xn satisfy � xi =1(allotherxj = 0) gives a natural deformation of Plancherel measure, studied for instance by =-=[ItTWi]-=-. Stanley [Sta] shows that this measure on partitions also arises by applying the Robinson-SchenstedKnuth algorithm to a random permutation distributed after a biased riffle shuffle (in other words, t... |

25 |
Some asymptotic results on finite vector spaces
- STONG
- 1988
(Show Context)
Citation Context ... the centralizer of the conjugacy class of GL(n, q) corresponding to the data {λφ} is 1 � φ qdeg(φ)·P i (λ′ φ,i )2 � i≥1 ( 1 q deg(φ) ) mi(λφ) The formulas given for conjugacy class size in [Kun] and =-=[St1]-=- are written in different form; for the reader’s benefit they have been expressed here in the form most useful to us. It follows that ⎡ ⎤ 1+ ∞� ZGL(n,q)u n = � ⎢ ⎣1+ � � n=1 φ�=z n≥1 λ⊢n xφ,λ . u n·de... |

25 |
Limit measures arising in the asymptotic theory of symmetric groups
- Vershik, Shmidt
- 1977
(Show Context)
Citation Context ...Ll] for results about cycle structure proved by a probabilistic interpretation of the cycle index generating function. Historically important papers in random permutation theory are [ErT], [Gon], and =-=[VeSc]-=-. Subsection 2.1 reviews the conjugacy classes of GL(n, q) and then discusses cycle indices for GL(n, q) andMat(n, q), the set of all n × n matrices with entries in the field of q elements. Subsection... |

23 | Coinvariants of nilpotent subalgebras of the Virasoro algebra and partition identities - Feigin, Frenkel |

23 | The boundary of Young lattice and random Young tableaux, Formal power series and algebraic combinatorics (New
- Kerov
- 1994
(Show Context)
Citation Context ... ;0, ) f 2 q µ KµΛ(0, 1/q). Next we give some background on potential theory on Bratteli diagrams. This is a beautiful subject, with connections to probability and representation theory. We recommend =-=[Ke1]-=- for background on potential theory with many examples and [BOl] for a survey of recent developments. The basic set-up is as follows. One starts with a Bratteli diagram, that is an oriented graded gra... |

22 | The Character of the infinite wedge representation
- Bloch, Okounkov
(Show Context)
Citation Context ...titions of size n gives a uniform partition. The measure Pq is related to the vertex operators [O1] and to the enumeration of ramified coverings of the torus [Dij]. In this regard the papers [O1] and =-=[BlO]-=- prove that the k point correlation function F (t1, ··· ,tk) = � q |λ| n� ∞� 1 λi−i+ 2 t λ k k=1 i=1 q j2 . .s76 JASON FULMAN is a sum of determinants involving genus 1 theta functions and their deriv... |

21 |
The probability of generating a finite simple group’, Geom. Dedicata 56
- Liebeck, Shalev
- 1995
(Show Context)
Citation Context ...r in a group G, one can easily see that 1 − P2,3(G) ≤ � M⊂G M maximal i2(M)i3(M) i2(G)i3(G) , since if x, y do not generate G, then they lie in some maximal subgroup M. Using this, Liebeck and Shalev =-=[LiSh]-=- show that if G is a simple classical group other than PSp4(q), then P2,3(G) → 1as|G| →∞. The paper [LiSh2] is another excellent example in which understanding the maximal subgroups of finite classica... |

21 | Simple groups, permutation groups, and probability
- Liebeck, Shalev
- 1999
(Show Context)
Citation Context ...y do not generate G, then they lie in some maximal subgroup M. Using this, Liebeck and Shalev [LiSh] show that if G is a simple classical group other than PSp4(q), then P2,3(G) → 1as|G| →∞. The paper =-=[LiSh2]-=- is another excellent example in which understanding the maximal subgroups of finite classical groups leads to powerful results. 3. Running example: General linear groups The purpose of this section i... |

21 | Generalized riffle shuffles and quasisymmetric functions
- Stanley
(Show Context)
Citation Context ...nt in representation theory and random matrix theory. Letting x1 = ··· = xn satisfy � xi =1(allotherxj = 0) gives a natural deformation of Plancherel measure, studied for instance by [ItTWi]. Stanley =-=[Sta]-=- shows that this measure on partitions also arises by applying the Robinson-SchenstedKnuth algorithm to a random permutation distributed after a biased riffle shuffle (in other words, this measure enc... |

20 |
The Cycle Structure of a Linear Transformation over a Finite Field
- Kung
- 1981
(Show Context)
Citation Context ...ual to λ in the sense that λ ′ i q )i denote (1 − u q Let n(λ) bethequantity � i≥1 (i − 1)λi and let ( u = mi(λ)+mi+1(λ)+···. u ) ···(1 − qi ). 2.1. The General Linear Groups. To begin we follow Kung =-=[Kun]-=- in defining a cycle index for GL(n, q). First it is necessary to understand the conjugacy classes of GL(n, q). As is explained in Chapter 6 of the textbook [Her], an element α ∈ GL(n, q) has its conj... |

19 | The rank of sparse random matrices over finite fields
- Blömer, Karp, et al.
- 1997
(Show Context)
Citation Context ... elements of GL(n, q) to be given in Subsection 3.1. Analogous universality results are known for matrices with complex entries [So]. For further information on the rank of random 0 − 1 matrices, see =-=[BKW]-=- for sparse matrices, [Bo] for a survey of results on the rank over the real numbers, and also the discussion of work of Rudvalis and Shinoda in Subsection 3.2. Elkies [El] studies the rank of Hankel ... |