Lattices from equiangular tight frames

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• 作者：Albrecht Bö ; ttcher^a ; ^{aboettch@mathematik.tu-chemnitz.de" class="auth_mail" title="E-mail the corresponding author} ; Lenny Fukshansky^b ; ^{lenny@cmc.edu" class="auth_mail" title="E-mail the corresponding author} ; Stephan Ramon Garcia^c ; ^{stephan.garcia@pomona.edu" class="auth_mail" title="E-mail the corresponding author} ; Hiren Maharaj^c ; ^{hirenmaharaj@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Deanna Needell^b ; ^{dneedell@cmc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 15B35 ; secondary ; 05B30 ; 11H06 ; 42C15 ; 52C07
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：510
• 期：Complete
• 页码：395-420
• 全文大小：474 K

文摘

We consider the set of all linear combinations with integer coefficients of the vectors of a unit equiangular tight <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516303974&_mathId=si1.gif&_user=111111111&_pii=S0024379516303974&_rdoc=1&_issn=00243795&md5=7a51ac883a9bccfe900886269649e322" title="Click to view the MathML source">(k,n)span><span class="mathContainer hidden"><span class="mathCode">$(k,n)$span>span>span> frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k  -dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516303974&_mathId=si2.gif&_user=111111111&_pii=S0024379516303974&_rdoc=1&_issn=00243795&md5=7fe3ecc9822197f71a60cf7b9cc05eb8" title="Click to view the MathML source">n=k+1span><span class="mathContainer hidden"><span class="mathCode">$n=k+1$span>span>span> and that there are infinitely many k   such that a lattice emerges for <span id="mmlsi173" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516303974&_mathId=si173.gif&_user=111111111&_pii=S0024379516303974&_rdoc=1&_issn=00243795&md5=6c7c07c7ae08e3bfd7187c41f039f353" title="Click to view the MathML source">n=2kspan><span class="mathContainer hidden"><span class="mathCode">$n=2k$span>span>span>. We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher's recent observation that this lattice is perfect.