Classifying Symmetric Spaces for SO(3,q) Jacob Sutter and Hanson Hao Illinois Mathematics and Science Academy Mentor: Dr. Ellen Ziliak, Benedictine University Japan Super Science Fair, November 15, 2018 Theorem 7. The number of unipotent matrices in the extended symmetric space R(3; p) for involution 1. Motivations 6. Involutions and Symmetric Spaces (1), where p is an odd prime, is: ( 2p − 1 p ≡ 1; 3 mod 8 Computer Graphics Rotational Matrices are also Special Orthogonal. Because special orthogonal jR j = • Our goal is to study symmetric spaces. First we must describe u matrices preserve size and shape, they are used to rotate objects in 3D space, and thus can be used 1 p ≡ 5; 7 mod 8 symmetry. by computers to animate objects in a 3D scene. Proof. For p ≡ 1; 3 mod 8, where −2 is a quadratic residue, take: p p 2 x2+8 −x2 −2+4x x2+4x −2 3 p8 8 p8 6−x2 −2−4x 4−x2 −x2 −2+4x7 2 3T 2 3 4 8p p4 8 5 ade ade x2−4x −2 −x2 −2−4x x2+8 4dbf 5 = 4dbf 5 8 8 8 efc efc where x ranges from 1 to p − 1 for p − 1 cases, p p 2 x2+8 x2 −2+4x x2−4x −23 p 8 8 p 8 2 2 2 T 6x −2−4x 4−x x −2+4x7 Quantum Physics The Special Orthogonal group is also related to the Special Unitary group, which • We can say a matrix is symmetric if B = B 4 8p p 4 8 5 have a basis formed by the three Pauli matrices. The Pauli matrices are used to represent spin of x2+4x −2 x2 −2−4x x2+8 • In the set of all symmetric matrices R = fB 2 jB = BT g is a symmetric space. 8 8 8 particles in quantum mechanics. R R • Notice that (BT )T = B where x ranges from 1 to p − 1 for another p − 1 cases, and the identity matrix for 1 case and a total • Our goal is to define a general version of this symmetric space. count of 2(p − 1) + 1 = 2p − 1. 2. Matrix Operations • Lets define another function θ(X) = AXA−1 that maps a matrix to another matrix. We observe that the same property holds, θ(θ(X)) = X for all X. For p ≡ 5; 7 mod 8, where −2 is not a quadratic residue, take the identity matrix for a total count of 1. • θ is called an involution, and A the involution matrix. A Definition 1. The Transpose Operator takes a matrix and flips it about its diagonal to form another • In this more general setting the symmetric space Theorem 8. The number of unipotent matrices in the extended symmetric space R(3; p) for involution AT −1 matrix, denoted . R = fX 2 S(3; q)jθ(X) = Xg (2), where p is an odd prime, is: T −1 T • Instead of a visual symmetry defined by B = B, θ(X) = X is a sort of algebraic symmetry. 2a b c3 2a d g3 ( 8 9 2p − 1 p ≡ 1 mod 4 4d e f5 = 4b e h5 20 0 43 20 1 03 21 1 33 jRuj = < = 1 p ≡ 3 mod 4 g h i c f i Example Elements in R(3; 5) : 40 4 05 ; 40 0 45 ; 43 4 45 : 4 0 0 4 0 0 1 2 1 ; Definition 2. The Determinant of a square matrix A is a notion of the area/volume/hyper-volume en- Proof. For p ≡ 1 mod 4, where −1 is a quadratic residue, take: closed within the region formed by the parallelpiped with sides being the row or column vectors of the matrix. p p • One way to create a symmetric matrix in R is to take any matrix C and multiply by it’s transpose. 2 2−c2 2−c2 3 T 2 2 −1 − −1 c We define a second symmetric space Q = fCC jC 2 g 2 p p 2 p R 2−c −1 − −1 c +2 c −1 4 2 p2 5 • It is important to note that Q ⊂ R −c −c −1 1 • In our more general notion of symmetry the set Q = fXθ(X)−1jX 2 SO(3; q)g where c ranges from 1 to p − 1 for p − 1 cases, 8 9 24 0 03 20 1 03 24 4 33 2 p 2 p < = 2 2−c −1 − 2−c −1 c 3 p 2 p 2 p Example Elements in Q(3; 5) : 40 1 05 ; 40 0 45 ; 41 2 15 2−c2 c2+2 4 −1 − −1 −c −15 : 0 0 4 4 0 0 3 4 4 ; 2 p2 −c c −1 1 • For the Special Orthogonal Group, we have four different types of maps θ that can be used to where c ranges from 1 to p − 1 for another p − 1 cases, and the identity matrix for 1 case and a total define an involution. count of 2(p − 1) + 1 = 2p − 1. 3. Modular Arithmetic Theorem 1 (Benim, Dometrius, Helminck, Wu). If θ(X) = AXA−1 is an involution then A2 = ±I. For p ≡ 3 mod 4, where −1 is not a quadratic residue, take the identity matrix for a total count of 1. For SO(3,q) we have 4 types of involutions to consider: Clock Arithmetic On a clock, 5:00 = 17:00 = 29:00 = ..., we call these elements congruent mod 12. A2 = I A2 = −I Type 1 Type 3 Conjecture 9. For all odd primes p, the number of unipotent matrices in the extended symmetric space But these times are placed together into a class, represented by 5:00. Fpp Fp[ α] Type 2 Type 4 R(3; p) for any type 1 involution and the number of unipotent matrices in the general symmetric space Q(3; p) for any type 1 involution are equal. • For Type 1 involutions, we define 2 isomorphic subclassses: – Class 1: Represented by 20 0 13 Ru = Qu (1) = 40 1 05 1 0 0 for all R and Q with given p. – Class 2: Represented by The following theorems and corollaries are for the other types of Involutions. 21 0 0 3 (2) = 40 1 0 5 Theorem 10. There are no Type 3 involutions for 3x3 matrices. 0 0 −1 Generalizing, we define congruence on the integers as follows: Proof. A Type 3 involution is θ(g) = AgA−1, where A2 ≡ −I and A is orthogonal. Thus, A has determi- • For Type 3 and 4 there are no involutions. Definition 3. Congruence. a ≡ b mod m if and only if m divides b − a. nant ±1 and A2 has determinant 1 mod p. However, • For Type 2 we only have preliminary data at this point and have not yet confirmed a count. Because congruent elements behave the same algebraically, we put them into a class and pick a single representative in place of enumerating them all. 2−1 0 0 3 −I = 4 0 −1 0 5 Definition 4. Congruence Class. For a given a 2 Z we can define a congruence class, written [a], 0 0 −1 such that [a] = fb 2 Zja ≡ b mod mg. We call a the representative of [a]. 7. Results has determinant −1. There is no odd p such that 1 ≡ −1 mod p, so there is no such A such that Theorem 2. The number of matrices in O(3; q), where q = pn and p is an odd prime, is twice the number A2 ≡ −I, and thus there are no Type 3 involutions for 3x3 matrices. 4. Special Orthogonal Group of matrices in SO(3; q): jO(3; q)j = 2 ∗ jSO(3; q)j: Corollary 11. There are no Type 3 involutions for any nxn matrices where n is odd. p p Proof. By previous results, an orthogonal matrix A in a finite field Fq has determinant ±1. Consider ma- ∼ Definition 5. A matrix A is special if the determinant of A is 1. Theorem 12. For all field extensions Fq[ α] =Fq[ β] where α and β are non-square in the finite field trix M which has determinant 1. We know that −M has determinant −1, but (−M)(−M)T = MM T = I, p ∼ p Fq, where q is a power of a prime. Therefore SO(3; q)[ α] = SO(3; q)[ β]. Definition 6. A matrix is orthogonal if all of its row (or column) vectors are pairwise orthogonal. Equiv- so −M is an orthogonal matrix. Thus there are the same number of matrices in O(3; p) with determi- T T alently, a matrix A is orthogonal if AA = A A = I. nant 1 as there are with determinant −1. As the matrices with determinant 1 make up SO(3; p), we Proof. Since α is non-square, 1 is also non-square. The product of two non-square values in a finite have that α field must be square (because their Legendre symbols, −1 and −1, multiply to 1), so β is square. Then p p p pα p let n ≡ pβ so that n α ≡ β. We can then create a correspondence between [ α] and [ β] by jO(3; p)j = 2 ∗ jSO(3; p)j: α p p Fq p Fq p substituting n α for β in the latter group. This correspondence is one-to-one as a + b β and c + d β cannot map to the same value unless a ≡ c and bn ≡ dn ) b ≡ d (since we are working in a finite p ∼ p Conjecture 3.