Cartography on unoriented surfaces, with applications to real and - PowerPoint PPT Presentation

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness This construction also works equally well for hypermaps, in which the hyperedge permutation must also satisfy α = δα − 1 δ . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness This construction also works equally well for hypermaps, in which the hyperedge permutation must also satisfy α = δα − 1 δ . 1 2 3 4 5 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness This construction also works equally well for hypermaps, in which the hyperedge permutation must also satisfy α = δα − 1 δ . 1 2 3 4 5 ϕ = (1 , 2 , 3) (4 , 5) ; Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness This construction also works equally well for hypermaps, in which the hyperedge permutation must also satisfy α = δα − 1 δ . 1 2 3 4 5 ϕ = (1 , 2 , 3) (4 , 5) ; α = (1 , − 3 , 4) ( − 4 , 3 , − 1) (2 , − 5) (5 , − 2) Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness This construction also works equally well for hypermaps, in which the hyperedge permutation must also satisfy α = δα − 1 δ . 1 2 3 4 5 ϕ = (1 , 2 , 3) (4 , 5) ; α = (1 , − 3 , 4) ( − 4 , 3 , − 1) (2 , − 5) (5 , − 2) σ = ϕ − 1 + α − 1 ϕ − = (1 , 5 , − 2 , 3 , − 4) (4 , − 3 , 2 , − 5 , − 1) Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Surfaces are classified as one of the following: Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Surfaces are classified as one of the following: ◮ spheres ( χ = 2), Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Surfaces are classified as one of the following: ◮ spheres ( χ = 2), ◮ n -holed tori ( χ = 0 , − 2 , − 4 , . . . ), Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Surfaces are classified as one of the following: ◮ spheres ( χ = 2), ◮ n -holed tori ( χ = 0 , − 2 , − 4 , . . . ), ◮ connected sums of n projective planes ( χ = 1 , 0 , − 1 , − 2 , − 3 , . . . ). Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Surfaces are classified as one of the following: ◮ spheres ( χ = 2), ◮ n -holed tori ( χ = 0 , − 2 , − 4 , . . . ), ◮ connected sums of n projective planes ( χ = 1 , 0 , − 1 , − 2 , − 3 , . . . ). The covering space of an orientable surface is two copies of the surface. Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Surfaces are classified as one of the following: ◮ spheres ( χ = 2), ◮ n -holed tori ( χ = 0 , − 2 , − 4 , . . . ), ◮ connected sums of n projective planes ( χ = 1 , 0 , − 1 , − 2 , − 3 , . . . ). The covering space of an orientable surface is two copies of the surface. The covering space of an unorientable surface is the orientable surface with Euler characteristic twice that of the original surface (so the connected sum of n projective planes is the ( n − 1)-holed torus). Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Definition Let I be a finite set of integers which does not contain both k and − k for any k . For a γ ∈ S ( I ) and a premap π ∈ PM ( ± I ), we define ϕ + ϕ − 1 ϕ − 1 + α − 1 ϕ − � � � � χ ( ϕ, α ) := # / 2 + # ( α ) / 2 + # / 2 − | I | . − Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Definition Let I be a finite set of integers which does not contain both k and − k for any k . For a γ ∈ S ( I ) and a premap π ∈ PM ( ± I ), we define ϕ + ϕ − 1 ϕ − 1 + α − 1 ϕ − � � � � χ ( ϕ, α ) := # / 2 + # ( α ) / 2 + # / 2 − | I | . − If ± I 1 and ± I 2 are disjoint, and γ i ∈ S ( I i ) and π i ∈ PM ( ± I i ) for i = 1 , 2, then χ ( γ 1 , π 1 ) + χ ( γ 2 , π 2 ) = χ ( γ 1 γ 2 , π 1 π 2 ) . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Theorem Let π, ρ ∈ S ( I ) for some finite set I. Then # ( π ) + # ( πρ ) + # ( ρ ) ≤ | I | + 2# � π, ρ � . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Theorem Let π, ρ ∈ S ( I ) for some finite set I. Then # ( π ) + # ( πρ ) + # ( ρ ) ≤ | I | + 2# � π, ρ � . Lemma Let ϕ ∈ S n , and let { V 1 , . . . , V r } ∈ P ( n ) be the orbits of ϕ . If α ∈ PM ( ± [ n ]) connects the blocks of {± V 1 , . . . , ± V r } , then χ ( γ, π ) ≤ 2 . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Orientable surfaces The Hyperoctahedral Group Unoriented surfaces The Quaternionic Case Classification and Euler characteristic Freeness Theorem Let π, ρ ∈ S ( I ) for some finite set I. Then # ( π ) + # ( πρ ) + # ( ρ ) ≤ | I | + 2# � π, ρ � . Lemma Let ϕ ∈ S n , and let { V 1 , . . . , V r } ∈ P ( n ) be the orbits of ϕ . If α ∈ PM ( ± [ n ]) connects the blocks of {± V 1 , . . . , ± V r } , then χ ( γ, π ) ≤ 2 . Surfaces with maximal Euler characteristic are typically associated with noncrossing diagrams. Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness In terms of indices, traces may be written X (1) i 1 i 2 X (2) i 2 i 3 · · · X ( n ) � Tr ( X 1 · · · X n ) = i n i 1 . 1 ≤ i 1 , i 2 ,..., i n ≤ N Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness In terms of indices, traces may be written X (1) i 1 i 2 X (2) i 2 i 3 · · · X ( n ) � Tr ( X 1 · · · X n ) = i n i 1 . 1 ≤ i 1 , i 2 ,..., i n ≤ N We can take traces along the cycles of a permutation π = ( c 1 , 1 , c 1 , 2 , . . . , c 1 , n 1 ) ( c 2 , 1 , . . . , c 2 , n 2 ) · · · ( c r , 1 , . . . , c r , n r ): � � Tr π ( X 1 , . . . , X n ) = Tr ( X 1 , 1 · · · X 1 , n 1 ) · · · Tr X c r , 1 · · · X c r , nr X (1) i 1 , i π (1) · · · X ( n ) � = i n , i π ( n ) . 1 ≤ i 1 ,..., i n Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Say we wish to calculate � � � � �� XY 1 XY 2 X T Y 3 XY 4 X T Y 5 X T Y 6 XY 7 XY 8 E tr tr . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Say we wish to calculate � � � � �� XY 1 XY 2 X T Y 3 XY 4 X T Y 5 X T Y 6 XY 7 XY 8 E tr tr . The traces of products are a sum over X i 1 j 1 Y (1) j 1 i 2 X i 2 j 2 Y (2) j 3 i 3 Y (3) i 3 i 4 X i 4 j 4 Y (4) j 5 i 5 Y (5) j 6 i 6 Y (6) i 6 i 7 X i 7 j 7 Y (7) j 7 i 8 X i 8 j 8 Y (8) j 2 j 3 X T j 4 j 5 X T i 5 i 1 X T j 8 j 6 . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness We construct the faces: Y 1 i 2 j 1 j 2 X Y 2 Y 6 X i 6 j 3 X T j 6 i 7 i 1 X X T Y 8 Y 5 j 8 j 7 X i 5 Y 7 i 8 i 3 X T Y 3 X j 5 i 4 Y 4 j 4 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness We use a result called the Wick formula. Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness We use a result called the Wick formula. There are three pairings on 4 elements: Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness We use a result called the Wick formula. There are three pairings on 4 elements: If X 1 , X 2 , X 3 , X 4 are components of a multivariate Gaussian random variable, then E ( X 1 X 2 X 3 X 4 ) = E ( X 1 X 2 ) E ( X 3 X 4 ) + E ( X 1 X 3 ) E ( X 2 X 4 ) + E ( X 1 X 4 ) E ( X 2 X 3 ) . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Let P 2 ( n ) be the set of pairings on n elements. Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Let P 2 ( n ) be the set of pairings on n elements. Theorem Let { f λ : λ ∈ Λ } , for some index set Λ , be a centred Gaussian family of random variables. Then for i 1 , . . . , i n ∈ Λ , � � E ( f i 1 · · · f i n ) = E ( f i k f i l ) . P 2 ( n ) { k , l }∈P 2 ( n ) Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Let P 2 ( n ) be the set of pairings on n elements. Theorem Let { f λ : λ ∈ Λ } , for some index set Λ , be a centred Gaussian family of random variables. Then for i 1 , . . . , i n ∈ Λ , � � E ( f i 1 · · · f i n ) = E ( f i k f i l ) . P 2 ( n ) { k , l }∈P 2 ( n ) Here, for a pairing π ∈ P 2 ( n ): � 1 , if i k = i l and j k = j l for all { k , l } ∈ π � E ( f i k j k f i l j l ) = . 0 , otherwise { k , l } Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Putting indices which must be equal next to each other, we get a surface gluing: Y 1 i 2 j 1 j 2 X Y 2 X Y 6 i 6 j 3 X T j 6 i 7 i 1 X X T Y 8 Y 5 j 8 j 7 X i 5 Y 7 i 8 i 3 X T Y 3 X j 5 i 4 Y 4 j 4 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness We note that if one term is from X and the other from X T , the edge identification is untwisted: Y 1 i 2 j 1 j 2 X Y 2 X Y 6 i 6 j 3 X T j 6 i 7 i 1 X X T Y 8 Y 5 j 8 j 7 X i 5 Y 7 i 8 i 3 X T Y 3 X T j 5 i 4 Y 4 j 4 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness If both terms are from X or from X T , the edge identification is twisted: Y 1 i 2 j 1 j 2 X Y 2 X Y 6 i 6 j 3 X T j 6 i 7 i 1 X X T Y 8 Y 5 j 8 j 7 X i 5 Y 7 i 8 i 3 X T Y 3 X T j 5 i 4 Y 4 j 4 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness The following vertex appears on the surface: i 3 i 6 i 4 Y T i 7 Y 6 3 i 1 i 2 Y 1 Y T 5 i 5 j 1 Y T j 7 7 i 8 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness The following vertex appears on the surface: i 3 i 6 i 4 Y T i 7 Y 6 3 i 1 i 2 Y 1 Y T 5 i 5 j 1 Y T j 7 7 i 8 If a corner appears upside-down, it is the transpose of that matrix which appears. Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness The following vertex appears on the surface: i 3 i 6 i 4 Y T i 7 Y 6 3 i 1 i 2 Y 1 Y T 5 i 5 j 1 Y T j 7 7 i 8 If a corner appears upside-down, it is the transpose of that matrix which appears. It contributes � � Y 1 Y T 3 Y 6 Y T 5 Y T Tr . 7 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness The same vertex viewed from the opposite side contributes the same value: i 3 i 3 i 6 i 6 i 4 i 4 Y T i 7 i 7 Y 3 Y T Y 6 3 6 i 2 i 1 i 1 i 2 Y T Y 1 Y T Y 5 5 1 i 5 i 5 j 1 j 1 Y T Y 7 j 7 7 i 8 i 8 j 7 � � � � Y 1 Y T 3 Y 6 Y T 5 Y T Y 7 Y 5 Y T 6 Y 3 Y T = Tr Tr . 7 1 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness 1 Let X : Ω → M M × N ( R ) be a random matrix with X ij = N f ij , √ where the f ij are independent N (0 , 1) random variables. Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness 1 Let X : Ω → M M × N ( R ) be a random matrix with X ij = N f ij , √ where the f ij are independent N (0 , 1) random variables. Definition Real Ginibre matrices are square matrices Z := X with M = N . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness 1 Let X : Ω → M M × N ( R ) be a random matrix with X ij = N f ij , √ where the f ij are independent N (0 , 1) random variables. Definition Real Ginibre matrices are square matrices Z := X with M = N . Definition Gaussian orthogonal ensemble matrices, or GOE matrices, are 1 � X + X T � symmetric matrices T := √ 2 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness 1 Let X : Ω → M M × N ( R ) be a random matrix with X ij = N f ij , √ where the f ij are independent N (0 , 1) random variables. Definition Real Ginibre matrices are square matrices Z := X with M = N . Definition Gaussian orthogonal ensemble matrices, or GOE matrices, are 1 � X + X T � symmetric matrices T := √ 2 Definition Real Wishart matrices are matrices W := X T D k X for some deterministic matrix D k . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness We wish to calculate expressions of the form � � �� X ( ε (1)) Y 1 · · · X ( ε ( n )) Y n E tr ϕ Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness We wish to calculate expressions of the form � � �� X ( ε (1)) Y 1 · · · X ( ε ( n )) Y n E tr ϕ � N − #( ϕ ) − n E � � = f ι 1 ι − 1 · · · f ι n ι − n � [ n ] → [ N ] ι : − [ n ] → [ M ] � � Y (1) ι − δε (1) ι δεϕ (1) · · · Y ( n ) E . ι − δε ( n ) ι δεϕ ( n ) Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness � � � � N − #( ϕ ) − n E Y (1) ι − δε (1) ι δεϕ (1) · · · Y ( n ) ι − δε ( n ) ι δεϕ ( n ) � π ∈P 2 ( n ) [ n ] → [ N ] ι : ι ± k = ι ± l : { k , l }∈ π − [ n ] → [ M ] Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness � � � � N − #( ϕ ) − n E Y (1) ι − δε (1) ι δεϕ (1) · · · Y ( n ) ι − δε ( n ) ι δεϕ ( n ) � π ∈P 2 ( n ) [ n ] → [ N ] ι : ι ± k = ι ± l : { k , l }∈ π − [ n ] → [ M ] Reversing the order of summation, � � � � N − #( ϕ ) − n E Y (1) ι − δε (1) ι δεϕ (1) · · · Y ( n ) ι − δε ( n ) ι δεϕ ( n ) π ∈P 2 ( n ) � [ n ] → [ N ] ι : − [ n ] → [ M ] ι ± k = ι ± l : { k , l }∈ π Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Regardless of the sign of k , we can write the entry Y ( k ) ι δδεϕ − ( k ) ι δεϕ +( k ) . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Regardless of the sign of k , we can write the entry Y ( k ) ι δδεϕ − ( k ) ι δεϕ +( k ) . The first index of Y ϕ − 1 − δ ε πδπδ ε ϕ + ( k ) is: ι δδ ε ϕ − ( ϕ − 1 − δ ε πδπδ ε ϕ + ( k ) ) = ι δπδπδ ε ϕ + ( k ) , which is equal to the second index of Y k . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Regardless of the sign of k , we can write the entry Y ( k ) ι δδεϕ − ( k ) ι δεϕ +( k ) . The first index of Y ϕ − 1 − δ ε πδπδ ε ϕ + ( k ) is: ι δδ ε ϕ − ( ϕ − 1 − δ ε πδπδ ε ϕ + ( k ) ) = ι δπδπδ ε ϕ + ( k ) , which is equal to the second index of Y k . N # ( ϕ − 1 � � − δ ε πδπδ ε ϕ + ) / 2 − #( ϕ ) − n E � tr FD ( ϕ − 1 − δ ε πδπδ ε ϕ + ) ( Y 1 , . . . , Y n ) . π ∈P 2 ( n ) Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Real Ginibre matrices are square matrices Z := X with M = N . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Real Ginibre matrices are square matrices Z := X with M = N . Thus � � �� Z ( ε (1)) Y 1 , . . . , Z ( ε ( n )) Y n E tr ϕ � � � N χ ( ϕ,δ ε αδ ε ) − #( ϕ ) E = tr FD ( ϕ − 1 − δ ε αδ ε ϕ + ) ( Y 1 , . . . , Y n ) . α ∈{ πδπ : π ∈P 2 ( n ) } Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Real Ginibre matrices are square matrices Z := X with M = N . Thus � � �� Z ( ε (1)) Y 1 , . . . , Z ( ε ( n )) Y n E tr ϕ � � � N χ ( ϕ,δ ε αδ ε ) − #( ϕ ) E = tr FD ( ϕ − 1 − δ ε αδ ε ϕ + ) ( Y 1 , . . . , Y n ) . α ∈{ πδπ : π ∈P 2 ( n ) } This is a sum over all gluings compatible with the edge directions given by the transposes. Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness 1 � X + X T � If we expand out the GOE matrix T := √ , we get 2 E ( tr ϕ ( TY 1 , . . . , TY n )) 1 � � �� � X ( ε (1)) Y 1 · · · X ( ε ( n )) Y n = 2 n / 2 E tr ϕ . ε : { 1 ,..., n }→{ 1 , − 1 } Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness If we collect terms, this is equivalent to summing over all edge-identifications. Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness If we collect terms, this is equivalent to summing over all edge-identifications. Thus E ( tr ϕ ( TY 1 , . . . , TY n )) � � � N χ ( ϕ,α ) − #( ϕ ) E = − αϕ + ) ( Y 1 , . . . , Y n ) tr FD ( ϕ − 1 . α ∈ PM ( ± [ n ]) ∩P 2 ( ± [ n ]) Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness With Wishart matrices W := X T D k X , we can collapse the edges corresponding to each matrix to a single edge. We can think of the connecting blocks as (possibly twisted) hyperedges. j 2 j 3 Y 1 i 2 X D 1 i 1 X T i 3 D 2 X T i 4 j 1 X Y 3 j 6 X j 4 X T Y 2 D 3 j 5 j 7 j 10 i 6 i 5 Y 5 X T X i 10 i 7 D 4 D 5 i 8 i 9 X X T Y 4 j 8 j 9 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness With Wishart matrices W := X T D k X , we can collapse the edges corresponding to each matrix to a single edge. We can think of the connecting blocks as (possibly twisted) hyperedges. j 2 j 3 Y 1 i 2 X D 1 i 1 X T i 3 D 2 X T i 4 j 1 X Y 3 j 6 X j 4 X T Y 2 D 3 j 5 j 7 j 10 i 6 i 5 Y 5 X T X i 10 i 7 D 4 D 5 i 8 i 9 X X T Y 4 j 8 j 9 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness With Wishart matrices W := X T D k X , we can collapse the edges corresponding to each matrix to a single edge. We can think of the connecting blocks as (possibly twisted) hyperedges. Y 1 W 1 1 W 2 Y 3 W 3 Y 2 Y 4 W 4 W 5 Y 5 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Thus: E ( tr ϕ ( W 1 Y 1 , · · · , W n Y n )) � N χ ( ϕ,α ) − #( ϕ ) tr FD ( α − 1 ) ( D 1 , . . . , D n ) = α ∈ PM ([ n ]) � � E tr FD ( ϕ − 1 − πϕ + ) ( Y 1 , . . . , Y n ) . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Definition A Haar-distributed orthogonal matrix is a random matrix with left-invariant probability measure on the orthogonal matrices. Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Definition A Haar-distributed orthogonal matrix is a random matrix with left-invariant probability measure on the orthogonal matrices. Theorem (Collins, ´ Sniady, 2006) � E ( O i 1 j 1 · · · O i n j n ) = Wg ( π 1 , π 2 ) ( π 1 ,π 2 ) ∈P 2 2 ( n ) i = i ◦ π 1 , j = j ◦ π 2 where: Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Definition A Haar-distributed orthogonal matrix is a random matrix with left-invariant probability measure on the orthogonal matrices. Theorem (Collins, ´ Sniady, 2006) � E ( O i 1 j 1 · · · O i n j n ) = Wg ( π 1 , π 2 ) ( π 1 ,π 2 ) ∈P 2 2 ( n ) i = i ◦ π 1 , j = j ◦ π 2 where: ◮ Wg ( π 1 , π 2 ) depends only on the block structure of π 1 ∨ π 2 ; Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Definition A Haar-distributed orthogonal matrix is a random matrix with left-invariant probability measure on the orthogonal matrices. Theorem (Collins, ´ Sniady, 2006) � E ( O i 1 j 1 · · · O i n j n ) = Wg ( π 1 , π 2 ) ( π 1 ,π 2 ) ∈P 2 2 ( n ) i = i ◦ π 1 , j = j ◦ π 2 where: ◮ Wg ( π 1 , π 2 ) depends only on the block structure of π 1 ∨ π 2 ; ◮ if π 1 ∨ π 2 has blocks 2 λ 1 , . . . , 2 λ s , then � s � ( − 1) λ k − 1 C λ k − 1 N − n � N − n 2 − s − 1 � � 2 − s + O Wg ( π 1 , π 2 ) = . k =1 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Say we wish to calculate � � � � �� OY 1 OY 2 O T Y 3 OY 4 O T Y 5 O T Y 6 OY 7 OY 8 E tr tr . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Say we wish to calculate � � � � �� OY 1 OY 2 O T Y 3 OY 4 O T Y 5 O T Y 6 OY 7 OY 8 E tr tr . � � O ι 1 ι − 1 Y (1) ι − 1 ι 2 O ι 2 ι − 2 Y (2) ι − 2 ι − 3 O T ι − 3 ι 3 Y (3) = E ι 3 ι 1 ι : ± [ n ] → [ N ] � × O ι 4 ι − 4 Y (4) ι − 5 ι 5 Y (5) ι − 6 ι 6 Y (6) ι 6 ι 7 O ι 7 ι − 7 Y (7) ι − 7 ι 8 O ι 8 ι − 8 Y (8) ι − 4 ι − 5 O T ι 5 ι − 6 O T ι − 8 ι 4 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness We construct the faces ι − 5 ι 5 ι − 4 Y 4 O T O Y 5 ι − 1 ι − 6 ι 4 Y 1 ι 1 O ι 2 Y 8 O T O Y 3 ι 6 ι − 8 ι − 2 ι 3 O T Y 6 O Y 2 O ι − 3 Y 7 ι 7 ι 8 ι − 7 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness � � � = O ι 1 ι − 1 · · · O ι 8 ι − 8 E ι : ± [ n ] → [ N ] � � Y (1) ι − 1 ι 2 Y (2) ι − 2 ι − 3 Y (3) ι 3 ι 1 Y (4) ι − 4 ι − 5 Y (5) ι 5 ι − 6 Y (6) ι 6 ι 7 Y (7) ι − 7 ι 8 Y (8) × E ι − 8 ι 4 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness � � � = O ι 1 ι − 1 · · · O ι 8 ι − 8 E ι : ± [ n ] → [ N ] � � Y (1) ι − 1 ι 2 Y (2) ι − 2 ι − 3 Y (3) ι 3 ι 1 Y (4) ι − 4 ι − 5 Y (5) ι 5 ι − 6 Y (6) ι 6 ι 7 Y (7) ι − 7 ι 8 Y (8) × E ι − 8 ι 4 � � � E O ι 1 ι − 1 · · · O ι 8 ι − 8 = Wg ( π + , π − ) ( π + ,π − ) ∈P 2 (8) 2 ι = ι ◦ δπ − δπ + Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Consider π + = (1 , 2) (3 , 5) (4 , 8) (6 , 7) and π − = (1 , 6) (2 , 5) (3 , 7) (4 , 8) . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness Consider π + = (1 , 2) (3 , 5) (4 , 8) (6 , 7) and π − = (1 , 6) (2 , 5) (3 , 7) (4 , 8) . ι − 5 ι 5 ι − 4 Y 4 O T O Y 5 ι − 1 ι − 6 ι 4 Y 1 ι 1 ι 2 O Y 8 O T O Y 3 ι 6 ι − 8 ι − 2 ι 3 O T Y 6 O Y 2 O ι − 3 Y 7 ι 7 ι 8 ι − 7 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness There are a number of vertices containing the Y k matrices. Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness There are a number of vertices containing the Y k matrices. ι 1 ι 2 Y 1 Y T 3 ι 3 ι − 1 Y 5 ι 5 ι − 6 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness There are a number of vertices containing the Y k matrices. ι 1 ι 2 Y 1 Y T 3 ι 3 ι − 1 Y 5 ι 5 ι − 6 This vertex contributes: � � Y 1 Y T Tr 3 Y 5 . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness There are also a number of vertices containing the O matrices. ι − 5 ι 5 ι − 4 Y 4 O T O Y 5 ι − 1 ι − 6 ι 4 Y 1 O ι 1 ι 2 O T Y 8 O Y 3 ι 6 ι − 8 ι − 2 ι 3 O T Y 6 Y 2 O O ι − 3 Y 7 ι 7 ι 8 ι − 7 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness There are also a number of vertices containing the O matrices. ι − 5 ι 5 ι − 4 Y 4 O T O Y 5 ι − 1 ι − 6 ι 4 Y 1 O ι 1 ι 2 O T Y 8 O Y 3 ι 6 ι − 8 ι − 2 ι 3 O T Y 6 Y 2 O O ι − 3 Y 7 ι 7 ι 8 ι − 7 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness There are also a number of vertices containing the O matrices. ι − 5 ι 5 ι − 4 Y 4 O T O Y 5 ι − 1 ι − 6 ι 4 Y 1 O ι 1 ι 2 O T Y 8 O Y 3 ι 6 ι − 8 ι − 2 ι 3 O T Y 6 Y 2 O O ι − 3 Y 7 ι 7 ι 8 ι − 7 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness We expect these to contribute: � v 1 2 , v 2 2 , . . . , v r � Wg 2 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness We expect these to contribute: � v 1 2 , v 2 2 , . . . , v r � Wg 2 � v 1 2 , v 2 2 , . . . , v r � = N r wg . 2 Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness The Y k vertices are given by: ϕ − 1 − δ ε π − δπ + δ ε ϕ + . Emily Redelmeier Cartography on unoriented surfaces

Cartography Random Matrices Combinatorics of traces The Hyperoctahedral Group Example The Quaternionic Case Matrix models Freeness The Y k vertices are given by: ϕ − 1 − δ ε π − δπ + δ ε ϕ + . The permutation π − δπ + = (1 , − 2 , 5 , − 3 , 7 , − 6) (6 , − 7 , 3 , − 5 , 2 , − 1) (4 , − 8) (8 , − 4) enumerates the points around the cycles of π + ∪ π − : π + 1 2 3 4 5 6 7 8 π − Emily Redelmeier Cartography on unoriented surfaces