On the Structure of Generalized Symmetric Spaces of SL n ( F q ) J. - PowerPoint PPT Presentation

On the Structure of Generalized Symmetric Spaces of SL n ( F q ) J. Schaefer with C. Buell, L. Helminck, V. Klima, C. Wright, and E. Ziliak Geometric Methods in Representation Theory The University of Missouri November 20, 2016 Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Motivation Real symmetric spaces were introduced by ´ E. Cartan as a special class of homogeneous Riemannian manifolds. Later generalized by M. Berger who gave classifications of the irreducible semisimple symmetric spaces. The goal of this talk is to explore the structure of generalized symmetric spaces for G = SL n ( k ) where k = F q . Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Previous Work Classified the involutions of SL 2 ( k ) where char( k ) � = 2. Described the extended symmetric space, R , and the generalized symmetric space, Q , related to SL 2 ( k ). Proved the following: Theorem Let k be a finite field of odd characteristic. Then R = Q for any involution of the group SL 2 ( k ) . Does this result extend to SL n ( k )? Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Outline Classify the involutions of SL n ( k ) where char( k ) � = 2. Determine the relationship between R and Q for two conjugacy classes of involutions of SL n ( k ). Provide the relationship between R and Q for the remaining conjugacy classes of involutions of SL n ( k ). Discuss the corresponding results when char( k ) = 2. Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

What are generalized symmetric spaces? Definition Let G be a group and θ ∈ Aut ( G ). Then θ is an involution if θ has order 2. Let G be a group and θ be an involution of G . Definition The fixed − point group is the set of elements given by H = { g ∈ G | θ ( g ) = g } . Definition The generalized symmetric space is the set G / H . Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

What are generalized symmetric spaces? τ : G → G given by τ ( g ) = g θ ( g ) − 1 τ induces an isomorphism of the coset space G / H onto τ ( G ) Generalized symmetric space G / H ∼ = { g θ ( g ) − 1 | g ∈ G } = Q Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

What are extended symmetric spaces? Let G be a group and θ be an involution of G . Definition The extended symmetric space is the set of elements given by R = { g ∈ G | θ ( g ) = g − 1 } . Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Relationship between R and Q In general, Q ⊆ R . g θ ( g ) − 1 � − 1 . � g θ ( g ) − 1 � = θ ( g ) g − 1 = � θ However, typically Q � = R . For example, consider the involution θ : SL 2 ( R ) → SL 2 ( R ) defined by θ ( A ) = ( A T ) − 1 . Then Q = { AA T | A ∈ SL 2 ( R ) } and R = { A ∈ SL 2 ( R ) | A = A T } . Clearly, Q ⊂ R but Q � = R . Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Main Tool: Twisted Conjugation SL n ( k ) acts on R by twisted conjugation: SL n ( k ) × R → R g . r = gr θ ( g ) − 1 Equivalence Relation on R: r 1 ∼ r 2 if and only if g . r 1 = r 2 for some g ∈ SL n ( k ) Orbit of r ∈ R : [ r ] = SL n ( k ) . r = { gr θ ( g ) − 1 | g ∈ SL n ( k ) } [ I n ] = SL n ( k ) . I n = { g θ ( g ) − 1 | g ∈ SL n ( k ) } = Q Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Main Tool: Twisted Conjugation The twisted conjugacy classes partition R R − Q = { r ∈ R | gr θ ( g ) − 1 � = I n for all g ∈ SL n ( k ) } Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

The involutions of SL n ( F q ) Three kinds of involutions: • Inner Involutions Inn x ( g ) = xgx − 1 { r ∈ R | g ( rx ) g − 1 � = x for all g ∈ SL n ( k ) } R − Q = = { r ∈ R | rx is not similar to x under SL n ( k ) } • Outer Involution θ ( g ) = g − T { r ∈ R | g ( r ) g T � = I n for all g ∈ SL n ( k ) } R − Q = = { r ∈ R | r is not congruent to I n under SL n ( k ) } • A composition of the two Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Inner involutions of SL n ( F q ) Theorem (Helminck, Wu, Dometrius) Let k be a finite field of odd characteristic and n > 2. The isomorphism classes for involutions of SL n ( k ) include Inn Y i where Y i is the block-diagonal matrix diag( I n − i , − I i ) for � � 1 , 2 , · · · , ⌈ n − 1 i ∈ 2 ⌉ . If n is odd, these are the only isomorphism classes of involutions. If n is even, there is one additional isomorphism class, namely, Inn L where �� 0 � 0 � 0 � � �� 1 1 1 L = diag , , . . . , 0 0 0 s p s p s p and s p is any non-square in the field. Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Inner involutions of SL n ( F q ) Recall for Inn x ( g ) = xgx − 1 { r ∈ R | g ( rx ) g − 1 � = x for all g ∈ SL n ( k ) } R − Q = = { r ∈ R | rx is not similar to x under SL n ( k ) } Tools: • Rational canonical form • (Waterhouse, 1984) If k is a finite field, then every matrix similar to a matrix A over k is actually similar to A over SL n ( k ), except for those cases in which there is an integer m such that (i) k ∗ � = ( k ∗ ) m and (ii) all the invariant factors of A are m th powers. Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Inner involutions of SL n ( F q ) Consider Inn Y i ( g ) = Y i gY − 1 where Y i = diag( I n − i , − I i ) for i � � 1 , 2 , . . . , ⌈ n − 1 i ∈ 2 ⌉ . � = X − 1 � X ∈ SL n ( k ) | Y i XY − 1 R = i � X ∈ SL n ( k ) | ( XY i ) 2 = I n � = = { X ∈ SL n ( k ) | minimal poly of XY i divides ( λ + 1)( λ − 1) } � � � XY i is similar to Y j = diag( I n − j , − I j ) over k � = X ∈ SL n ( k ) � for some j in { 0 , 1 , 2 , . . . , n } � � � X ∈ SL n ( k ) | X = PY i P − 1 Y − 1 = for some P ∈ SL n ( k ) Q i = { X ∈ SL n ( k ) | XY i is similar to Y i over SL n ( k ) } = { X ∈ SL n ( k ) | XY i is similar to Y i over k } Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Inner involutions of SL n ( F q ) � � � XY i is similar to Y j over k for some j � R − Q = X ∈ SL n ( k ) � in { 0 , 1 , 2 , . . . , n } with j � = i � � � � AY − 1 = | A ∈ cl ( Y j ) . i j ∈{ 0 , 1 ,..., n }−{ i } j ≡ i mod 2 Example SL 3 ( k ) has only one inner involution, Inn Y 1 . = { X ∈ SL 3 ( k ) | XY 1 is similar to Y 1 or Y 3 over k } and R Q = { X ∈ SL 3 ( k ) | XY 1 is similar to Y 1 over k } . � � R − Q = { AY − 1 − I 3 Y − 1 | A ∈ cl ( Y 3 ) } = = { diag( − 1 , − 1 , 1) } 1 1 as cl ( Y 3 ) = cl ( − I 3 ) = {− I 3 } . Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Outer involutions of SL n ( F q ) Theorem (Helminck, Wu, Dometrius) Let k be a finite field of odd characteristic, s p be a representative of the non-square class of k ∗ / ( k ∗ ) 2 , I j be the j × j identity matrix, Inn G represent conjugation by a matrix G , and M be the matrix � I n − 1 � 0 for an integer n > 2. Furthermore, for even n define 0 s p � � 0 I n / 2 the matrix J as . If n is odd, then there are two − I n / 2 0 isomorphism classes of outer involutions for SL n ( k ); representatives are θ 1 given by θ 1 ( X ) = X − T and θ 1 ◦ Inn M . If n is even, then there are three isomorphism classes of outer involutions for SL n ( k ); representatives are given by θ 1 , θ 1 ◦ Inn M , and θ 1 ◦ Inn J . Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Outer Involutions of SL n ( k ) Consider the outer involution θ 1 with θ 1 ( g ) = g − T . { r ∈ R | g ( r ) g T � = I n for all g ∈ SL n ( k ) } R − Q = = { r ∈ R | r is not congruent to I n over SL n ( k ) } Tools: • (Albert, 1938) For k be a finite field of odd characteristic and A ∈ GL n ( k ). (1) If A is symmetric, then A is congruent over k to a diagonal matrix. (2) If n = 2 ℓ and A is skew-symmetric, then A is congruent � 0 � I ℓ over k to the block matrix J = . − I ℓ 0 • (BHKSWZ) Every diagonal matrix in SL n ( k ) is congruent to I n over SL n ( k ). Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

Outer Involutions of SL n ( k ) θ 1 with θ 1 ( g ) = g − T { r ∈ R | g ( r ) g T � = I n for all g ∈ SL n ( k ) } R − Q = = { r ∈ R | r is not congruent to I n over SL n ( k ) } What do matrices in R look like? R = { r ∈ SL n ( k ) | r − T = r − 1 } = { r ∈ SL n ( k ) | r T = r } • By Albert r is congruent to a diagonal matrix d • By BHKSWZ d is congruent to I n Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )

The symmetric spaces of SL n ( F q ) Involution Result θ 1 ( X ) = X − T R = Q θ 1 ◦ Inn M R = Q θ 1 ◦ Inn J R � = Q , | R | = 2 | Q | Inn L R = Q Inn Y i R � = Q Schaefer with Buell, Helminck, Klima, Wright, and Ziliak Symmetric Spaces of SL n ( F q )