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Lattices from Group Frames and Vertex Transitive Graphs

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Lattices from Group Frames and Vertex Transitive Graphs Frames Lattices Graphs Examples Lattices from group frames and vertex transitive graphs Lenny Fukshansky Claremont McKenna College (joint work with Deanna Needell, Josiah Park and Jessie Xin) Tight frame F is rational if there exists a real number α so that αpf i ; f j q P Q @ 1 ¤ i; j ¤ n: Frames Lattices Graphs Examples Tight frames k A spanning set tf 1;:::; f nu Ă R , n ¥ k, is called a tight frame k if there exists a real constant γ such that for every x P R , n 2 2 }x} “ γ px; f j q ; j“1 ¸ where p ; q stands for the usual dot-product. Frames Lattices Graphs Examples Tight frames k A spanning set tf 1;:::; f nu Ă R , n ¥ k, is called a tight frame k if there exists a real constant γ such that for every x P R , n 2 2 }x} “ γ px; f j q ; j“1 ¸ where p ; q stands for the usual dot-product. Tight frame F is rational if there exists a real number α so that αpf i ; f j q P Q @ 1 ¤ i; j ¤ n: k Let f P R be a nonzero vector, then Gf “ tUf : U P Gu k is called a group frame (or G-frame). If G acts irreducibly on R , Gf is called an irreducible group frame. Irreducible group frames are always tight. Frames Lattices Graphs Examples Group frames Let G be a finite subgroup of Ok pRq, the k-dimensional real orthog- k onal group, then G acts on R by left matrix multiplication. This action is irreducible if it has no invariant subspaces except for t0u k and R . Frames Lattices Graphs Examples Group frames Let G be a finite subgroup of Ok pRq, the k-dimensional real orthog- k onal group, then G acts on R by left matrix multiplication. This action is irreducible if it has no invariant subspaces except for t0u k and R . k Let f P R be a nonzero vector, then Gf “ tUf : U P Gu k is called a group frame (or G-frame). If G acts irreducibly on R , Gf is called an irreducible group frame. Irreducible group frames are always tight. k When is LpFq a lattice, i.e. a discrete subgroup of R ? Theorem 1 (B¨ottcher,F., Needell, Park, Xin (2017/2018)) k The set LpFq is a full-rank lattice in R if and only if the tight frame F is rational. Remark: As we recently learned, a similar result is proved in a 2017 paper by T. Sunada. What kind of lattices can we get this way? What kind of lattices do we want to get? Frames Lattices Graphs Examples Tight frames and lattices Let F “ tf 1;:::; f nu be a tight pn; kq-frame and define n LpFq :“ spanZ F “ ai f i : a1;:::; an P Z : #i“1 + ¸ Theorem 1 (B¨ottcher,F., Needell, Park, Xin (2017/2018)) k The set LpFq is a full-rank lattice in R if and only if the tight frame F is rational. Remark: As we recently learned, a similar result is proved in a 2017 paper by T. Sunada. What kind of lattices can we get this way? What kind of lattices do we want to get? Frames Lattices Graphs Examples Tight frames and lattices Let F “ tf 1;:::; f nu be a tight pn; kq-frame and define n LpFq :“ spanZ F “ ai f i : a1;:::; an P Z : #i“1 + ¸ k When is LpFq a lattice, i.e. a discrete subgroup of R ? Remark: As we recently learned, a similar result is proved in a 2017 paper by T. Sunada. What kind of lattices can we get this way? What kind of lattices do we want to get? Frames Lattices Graphs Examples Tight frames and lattices Let F “ tf 1;:::; f nu be a tight pn; kq-frame and define n LpFq :“ spanZ F “ ai f i : a1;:::; an P Z : #i“1 + ¸ k When is LpFq a lattice, i.e. a discrete subgroup of R ? Theorem 1 (B¨ottcher,F., Needell, Park, Xin (2017/2018)) k The set LpFq is a full-rank lattice in R if and only if the tight frame F is rational. What kind of lattices can we get this way? What kind of lattices do we want to get? Frames Lattices Graphs Examples Tight frames and lattices Let F “ tf 1;:::; f nu be a tight pn; kq-frame and define n LpFq :“ spanZ F “ ai f i : a1;:::; an P Z : #i“1 + ¸ k When is LpFq a lattice, i.e. a discrete subgroup of R ? Theorem 1 (B¨ottcher,F., Needell, Park, Xin (2017/2018)) k The set LpFq is a full-rank lattice in R if and only if the tight frame F is rational. Remark: As we recently learned, a similar result is proved in a 2017 paper by T. Sunada. What kind of lattices do we want to get? Frames Lattices Graphs Examples Tight frames and lattices Let F “ tf 1;:::; f nu be a tight pn; kq-frame and define n LpFq :“ spanZ F “ ai f i : a1;:::; an P Z : #i“1 + ¸ k When is LpFq a lattice, i.e. a discrete subgroup of R ? Theorem 1 (B¨ottcher,F., Needell, Park, Xin (2017/2018)) k The set LpFq is a full-rank lattice in R if and only if the tight frame F is rational. Remark: As we recently learned, a similar result is proved in a 2017 paper by T. Sunada. What kind of lattices can we get this way? Frames Lattices Graphs Examples Tight frames and lattices Let F “ tf 1;:::; f nu be a tight pn; kq-frame and define n LpFq :“ spanZ F “ ai f i : a1;:::; an P Z : #i“1 + ¸ k When is LpFq a lattice, i.e. a discrete subgroup of R ? Theorem 1 (B¨ottcher,F., Needell, Park, Xin (2017/2018)) k The set LpFq is a full-rank lattice in R if and only if the tight frame F is rational. Remark: As we recently learned, a similar result is proved in a 2017 paper by T. Sunada. What kind of lattices can we get this way? What kind of lattices do we want to get? A lattice L is well-rounded (WR) if spanR L “ spanR SpLq: Frames Lattices Graphs Examples Some properties of lattices Minimal norm of a lattice L is |L| “ min t}x} : x P Lzt0uu ; where }} is Euclidean norm. The set of minimal vectors of L is SpLq “ tx P L : }x} “ |L|u : From here on, let us write SpLq “ tx1;:::; xnu, where n, the cardi- nality of the set of minimal vectors of L must be even, and can be smaller or larger than (twice) the rank of the lattice. For example k SpZ q “ t˘e1;:::; ˘ek u ; so n “ 2k. Frames Lattices Graphs Examples Some properties of lattices Minimal norm of a lattice L is |L| “ min t}x} : x P Lzt0uu ; where }} is Euclidean norm. The set of minimal vectors of L is SpLq “ tx P L : }x} “ |L|u : From here on, let us write SpLq “ tx1;:::; xnu, where n, the cardi- nality of the set of minimal vectors of L must be even, and can be smaller or larger than (twice) the rank of the lattice. For example k SpZ q “ t˘e1;:::; ˘ek u ; so n “ 2k. A lattice L is well-rounded (WR) if spanR L “ spanR SpLq: A lattice L is called perfect if the set of symmetric matrices J txi xi : xi P SpLqu spans the space of k ˆ k symmetric matrices. Both, eutactic and perfect lattices are necessarily well-rounded. Up to similarity, there are only finitely many eutactic or perfect lattices in a given dimension. Frames Lattices Graphs Examples Eutaxy and perfection k A lattice L Ă R is called eutactic if there exist positive real num- bers c1;:::; cn such that n 2 2 }v} “ ci pv; xi q i“1 ¸ k for every vector v P R . If c1 “ ¨ ¨ ¨ “ cn, we say that L is strongly eutactic. Both, eutactic and perfect lattices are necessarily well-rounded. Up to similarity, there are only finitely many eutactic or perfect lattices in a given dimension. Frames Lattices Graphs Examples Eutaxy and perfection k A lattice L Ă R is called eutactic if there exist positive real num- bers c1;:::; cn such that n 2 2 }v} “ ci pv; xi q i“1 ¸ k for every vector v P R . If c1 “ ¨ ¨ ¨ “ cn, we say that L is strongly eutactic. A lattice L is called perfect if the set of symmetric matrices J txi xi : xi P SpLqu spans the space of k ˆ k symmetric matrices. Frames Lattices Graphs Examples Eutaxy and perfection k A lattice L Ă R is called eutactic if there exist positive real num- bers c1;:::; cn such that n 2 2 }v} “ ci pv; xi q i“1 ¸ k for every vector v P R . If c1 “ ¨ ¨ ¨ “ cn, we say that L is strongly eutactic. A lattice L is called perfect if the set of symmetric matrices J txi xi : xi P SpLqu spans the space of k ˆ k symmetric matrices. Both, eutactic and perfect lattices are necessarily well-rounded. Up to similarity, there are only finitely many eutactic or perfect lattices in a given dimension. k Space of full-rank lattices in R is identified with GLk pRq{ GLk pZq, and δ is a continuous function on this space.
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