Characterizations of symmetric cones by means of the basic relative - PowerPoint PPT Presentation

Characterizations of symmetric cones by means of the basic relative invariants Hideto Nakashima Kyushu University 2015/6/24 RIMS, Kyoto university Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 1 / 23

Background V = Sym( r, R ) , Ω = Sym( r, R ) ++ , W = V C (= Sym( r, C )) ∆ 1 ( w ) , . . . , ∆ r ( w ): the principal minors of w ∈ W T Ω := Ω + iV : Tube domain Classical fact Put ∆ 0 ( w ) = 1 . If w ∈ T Ω , then one has ∆ k ( w ) Re ∆ k − 1 ( w ) > 0 ( k = 1 , . . . , r ) . → This result can be generalize to any irreducible symmetric cone. (Ishi–Nomura 2008) Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 2 / 23

Background V : simple Euclidean Jordan algebra Ω: irreducible symmetric cone of V T Ω := Ω + iV ⊂ W = V C ∆ 1 ( x ) , . . . , ∆ r ( x ): the principal minors of V → naturally continued to holomorphic polynomial functions of W Theorem A (Ishi–Nomura 2008) Put ∆ 0 ( w ) = 1 . If w ∈ T Ω , then one has ∆ k ( w ) Re ∆ k − 1 ( w ) > 0 ( k = 1 , . . . , r ) . Q. Does this property characterize symmetric cones? A. No (Ishi-Nomura 2008) Q. How does this property generalize to homogeneous cones? → Today’s topic Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 3 / 23

Talking plan (1) Background (i) Theorem A (2) Generalization of Theorem A (i) Setting and definitions (ii) matrix realization of homogeneous cones (iii) known results (iv) Theorem 1 (generalization of Theorem A) (3) Characterization of symmetric cones (i) dual cones (ii) Main theorem (characterization of symmetric cones) (iii) sketch of the proof Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 4 / 23

Setting V : finite-dimensional real vector space Ω: open convex cone in V containing no entire line G (Ω) := { g ∈ GL ( V ); g (Ω) = Ω } Ω is homogeneous ⇔ G (Ω) acts on Ω transitively Assume that Ω is homogeneous ∃ H : split solvable Lie subgroup of G (Ω) s.t. H � Ω: simply transitively. Example S N = Sym( N, R ) N = Sym( N, R ) ++ = { x ∈ V ; x is positive definite } S + g ∈ GL ( N, R ) acts on S + N by g · x := gx t g . H N : group of lower triangular matrices with positive diagonals. → H N acts on S + N simply transitively Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 5 / 23

Matrix realization of homogeneous cones (Ishi 2006) N = n 1 + · · · + n r : partition of N ∈ N V lk ⊂ Mat( n l , n k ; R ): system of vector spaces satisfying (V0) V jj = R I n j ( j = 1 , . . . , r ) , (V1) A ∈ V lk , B ∈ V kj ⇒ AB ∈ V lj ( j < k < l ) , (V2) A ∈ V lj , B ∈ V kj ⇒ A t B ∈ V lk ( j < k < l ) , (V3) A ∈ V kj ⇒ A t A ∈ V kk ( j < k ) .   X 11 t X 21 · · · t X r 1       ... X kk = x kk I n k ,   t X r 2     X 21 X 22     Z V = X = ; ( x kk ∈ R ) ⊂ S N , .  ...  .   . X lk ∈ V lk           X r 1 X r 2 · · · X rr   P V = { X ∈ Z V ; X is positive definite } . → P V is a homogeneous cone of rank r . . . Any homogeneous cone Ω can be realized as some P V . Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 6 / 23

Split solvable Lie subgroup H H is linearly isomorphic to     T 11  T kk = e t k / 2 I n k    T 21 T 22       h =  ; ( t k ∈ R ) ⊂ H N . .  ...  .   . T lk ∈ V lk        T r 1 T r 2 · · · T rr   The action on P V is described as h · x = hx t h . Define. f : relatively H -invariant function of Ω ∃ χ : H → R : 1-dim. rep. s.t. f ( h · x ) = χ ( h ) f ( x ) . → ∃ ν = ( ν 1 , . . . , ν r ) ∈ R r s.t. χ ( h ) = e ν 1 t 1 + ··· + ν r t r (multiplier). In particular we have f (diag( x 1 , . . . , x r )) = x ν 1 1 · · · x ν r r Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 7 / 23

Basic relative invariants Theorem (Ishi–Nomura 2008) There exist just r relatively H -invariant irreducible polynomials ∆ 1 ( x ) , . . . , ∆ r ( x ) , and Ω is described as Ω = { x ∈ V ; ∆ 1 ( x ) > 0 , . . . , ∆ r ( x ) > 0 } . → ∆ j ’s are called the basic relative invariants of Ω . σ j = ( σ j 1 , . . . , σ jr ): multiplier of ∆ j ( x )   σ 1 . . σ :=  = ( σ jk ) 1 ≤ j,k ≤ r : multiplier matrix   .  σ r multiplier matrix is lower, the diagonal elements are all 1 (Ishi 2001). We have an algorithm for calculating σ (N-. 2014). Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 8 / 23

Complexification W := V C , and T Ω := Ω + iV . H C : complexification of H . f C : complexification of a relatively H -invariant function f . relatively H C -invariance “ f C ( ρ ( h ) w ) = χ ( h ) f C ( w ) for ∀ h ∈ H C , w ∈ W ” → ∃ h ′ s.t. ρ ( h ) w = ρ ( h ′ ) w for ∀ w ∈ W , but χ ( h ) ? = χ ( h ′ ) . . . If f is rational, then χ is well-defined. ∆ 1 , . . . , ∆ r : naturally continued to holomorphic poly. S := { w ∈ W ; ∃ ∆ j ( w ) = 0 } . { } Put N C := h ∈ H C ; diag = I n j ( j = 1 , . . . , r ) . f C ( n · w ) = f C ( w ) , Then f C (diag( x 1 , . . . , x r )) = x ν 1 1 · · · x ν r r . Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 9 / 23

Known results Proposition (Ishi-Nomura 2008) (i) For any w ∈ W \S , there exist unique n ∈ N C and α j ( w ) ∈ C × ( j = 1 , . . . , r ) such that w = n · diag( α 1 ( w ) , . . . , α r ( w )) . (ii) If w ∈ T Ω , then one has Re α k ( w ) > 0 for k = 1 , . . . , r . → Describe α 1 ( w ) , . . . , α r ( w ) by using ∆ 1 ( w ) , . . . , ∆ r ( w ) . For µ, τ ∈ Z r , put α µ ( w ) := α 1 ( w ) µ 1 · · · α r ( w ) µ r , ∆ τ ( w ) := ∆ 1 ( w ) τ 1 · · · ∆ r ( w ) τ r , j ˇ e j := (0 , . . . , 0 , 1 , 0 , . . . , 0) . Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 10 / 23

Generalization of Theorem A Theorem 1 Let w ∈ T Ω . Then one has α j ( w ) = ∆ e j σ − 1 ( w ) , and hence Re ∆ e j σ − 1 ( w ) > 0 ( j = 1 , . . . , r ) . proof. For each j , we have ∆ j ( w ) = ∆ j ( n · diag( α 1 ( w ) , . . . , α r ( w ))) = ∆ j (diag( α 1 ( w ) , . . . , α r ( w ))) = α 1 ( w ) σ j 1 · · · α r ( w ) σ jr = α σ j ( w ) . → ∆ τ ( w ) = ( α σ 1 ( w )) τ 1 · · · ( α σ r ( w )) τ r = α τσ ( w ) . α j ( w ) = α e j ( w ) = ∆ e j σ − 1 ( w ) . Thus we have Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 11 / 23

Case of symmetric cones V : Euclidean Jordan algebra ∆ j ( x ): principal minors of V j In this case we have σ j = (1 , . . . , 1 , 0 , . . . , 0) and hence     1 0 1 0 1 1 − 1 1      → σ − 1 = σ =  .    ... ... ... ... .     .    1 · · · 1 1 0 − 1 1 Thus Theorem 1 leads us to the known result: If w ∈ Ω + iV , then one has Re ∆ e j σ − 1 ( w ) = Re ∆ j − 1 ( w ) − 1 ∆ j ( w ) ∆ j ( w ) = Re ∆ j − 1 ( w ) > 0 . Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 12 / 23

Talking plan (1) Background (i) Theorem A (2) Generalization of Theorem A (i) Setting and definitions (ii) matrix realization of homogeneous cones (iii) known results (iv) Theorem 1 (generalization of Theorem A) (3) Characterization of symmetric cones (i) dual cones (ii) Main theorem (characterization of symmetric cones) (iii) sketch of the proof Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 13 / 23

Dual cone Ω: homogeneous cone in V ⟨·|·⟩ : inner product of V Dual cone Ω ∗ of Ω is defined to be Ω ∗ := { } x ∈ V ; ⟨ x | y ⟩ > 0 for all y ∈ Ω \{ 0 } . ∆ ∗ 1 ( x ) , . . . , ∆ ∗ r ( x ): basic relative invariants of Ω ∗ the index is determined as the multiplier matrix σ ∗ to be upper triangular Ω is irreducible ⇔ Ω = Ω 1 ⊕ Ω 2 implies Ω 1 = { 0 } or Ω 2 = { 0 } . Theorem (Yamasaki 2014) Let Ω be an irreducible homogeneous cone. Then Ω is symmetric if and only if { deg ∆ 1 , . . . , deg ∆ r } = { deg ∆ ∗ 1 , . . . , deg ∆ ∗ r } = { 1 , . . . , r } . Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 14 / 23

Example V = S 3 3 and Ω ∗ = S + Ω = S + 3 (symmetric cone)     x 1 x 21 x 31    ; x i , x kj ∈ R V =  x = x 21 x 2 x 32  x 31 x 32 x 3  The basic relative invariants are described as ∆ ∗ ∆ 1 ( x ) = x 1 , 1 ( x ) = det x, ∆ 2 ( x ) = x 1 x 2 − x 2 ∆ ∗ 2 ( x ) = x 3 x 2 − x 2 21 , 32 , ∆ ∗ ∆ 3 ( x ) = det x, 3 ( x ) = x 3 . The multiplier matrices is given as     1 0 0 1 1 1  ,  . σ = 1 1 0 σ ∗ = 0 1 1   1 1 1 0 0 1 Hideto Nakashima (Kyushu Univ.) Characterizations of symmetric cones 2015/6/24 15 / 23