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{"Title": "Orders of the automorphism groups of the classical lattices", "Definition": "For a Euclidean lattice $L$ of dimension $n$ with Gram matrix $G$, the order $|\\operatorname{Aut}(L)|$ of its full linear isometry group is $\\#\\{U\\in\\operatorname{GL}_n(\\mathbb{Z}):U^{\\mathsf T}GU=G\\}$.", "Parameters": {"family": {"title": "family of the lattice", "type": "Symbolic", "values": {"Z": "$\\mathbb{Z}^n$", "A": "$A_n$", "A*": "$A_n^{*}$", "D": "$D_n$", "D*": "$D_n^{*}$", "E": "$E_n$", "E*": "$E_n^{*}$", "Lambda": "$\\Lambda_n$", "K": "$K_n$"}, "display": "family"}, "n": {"constraints": "$n\\geq 1$ for $\\mathbb{Z}^n$ and $A_n$; $n\\geq 4$ for $D_n$; $n\\in\\{6,7,8\\}$ for $E_n$; $n\\geq 3$ for $A_n^{*}$; $n\\geq 5$ for $D_n^{*}$; $n\\in\\{6,7\\}$ for $E_n^{*}$; $n\\geq 9$ for $\\Lambda_n$; $n=12$ for $K_n$", "display": "$n$", "title": "dimension", "type": "Z"}}, "Tags": ["discrete geometry", "number theory", "groups"], "Links": {"Catalogue": {"title": "G. Nebe and N. J. A. Sloane, A Catalogue of Lattices", "url": "https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/"}, "PARI": {"title": "PARI/GP reference: qfauto, lattice automorphism groups", "url": "https://pari.math.u-bordeaux.fr/dochtml/html-stable/Linear_algebra__Vectors__matrices_and_sets.html#qfauto"}, "OEISBW": {"title": "OEIS A014116: orders of automorphism groups of Barnes-Wall lattices", "url": "https://oeis.org/A014116"}}, "Data properties": {"type": "Z", "rigour": "exact", "complete": "no", "complete-note": "it holds all 133 lattices of HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices}[the classical-lattice packing table] through dimension $24$, including every listed dimension of each family; dimension $24$ includes the Leech lattice and provides a common endpoint for comparison with the packing table", "rigour details": "Every value is an exact integer. The cubic and root-lattice orders are computed by integer products, with the diagram-symmetry factors included; duals use the same orders. The laminated and Coxeter-Todd orders are computed from their integral catalogue Gram matrices by PARI qfauto CITE{PARI}, implementing the Plesken-Souvignier algorithm CITE{PleskenSouvignier}. The attached generator checks that each returned matrix has determinant $\\pm1$ and preserves the Gram matrix.\n\nIndependent Gram-matrix computations verify all 116 cubic, root and dual entries, with known controls of orders $8$ and $12$ for the square and hexagonal lattices. Contragredient matrices preserve the inverse Gram forms for all 44 listed duals. All 17 catalogue matrices agree entry by entry with the published catalogue and with the attachment of the classical-lattice packing table. The 26 applicable stored Coxeter Poincar\u00e9 polynomials evaluated at $1$ agree with the root and cubic orders after the diagram-symmetry factors. Further checks compare the exceptional root orders and the Leech order with the catalogue, the Barnes-Wall order with OEIS A014116, the real Coxeter-Todd order with the index-two relation on page $429$ of CITE{CoxeterTodd}, and the dimension-$23$ and dimension-$24$ orders with twice the stored orders of $\\mathrm{Co}_2$ and $\\mathrm{Co}_1$. The Coxeter-Todd source has a printed final factor $8$ where its index-two argument and independent Gram computation give $7$."}, "Display properties": {"number-header": "$|\\operatorname{Aut}(L)|$"}, "Numbers": {"Z": {"1": "2", "2": {"number": "8", "comment": "The square lattice."}, "3": "48", "4": "384", "5": "3840", "6": "46080", "7": "645120", "8": "10321920", "9": "185794560", "10": "3715891200", "11": "81749606400", "12": "1961990553600", "13": "51011754393600", "14": "1428329123020800", "15": "42849873690624000", "16": "1371195958099968000", "17": "46620662575398912000", "18": "1678343852714360832000", "19": "63777066403145711616000", "20": "2551082656125828464640000", "21": "107145471557284795514880000", "22": "4714400748520531002654720000", "23": "216862434431944426122117120000", "24": "10409396852733332453861621760000"}, "A": {"1": "2", "2": {"number": "12", "comment": "The hexagonal lattice."}, "3": {"number": "48", "comment": "The face-centred cubic lattice."}, "4": "240", "5": "1440", "6": "10080", "7": "80640", "8": "725760", "9": "7257600", "10": "79833600", "11": "958003200", "12": "12454041600", "13": "174356582400", "14": "2615348736000", "15": "41845579776000", "16": "711374856192000", "17": "12804747411456000", "18": "243290200817664000", "19": "4865804016353280000", "20": "102181884343418880000", "21": "2248001455555215360000", "22": "51704033477769953280000", "23": "1240896803466478878720000", "24": "31022420086661971968000000"}, "A*": {"3": {"number": "48", "comment": "The body-centred cubic lattice."}, "4": "240", "5": "1440", "6": "10080", "7": "80640", "8": "725760", "9": "7257600", "10": "79833600", "11": "958003200", "12": "12454041600", "13": "174356582400", "14": "2615348736000", "15": "41845579776000", "16": "711374856192000", "17": "12804747411456000", "18": "243290200817664000", "19": "4865804016353280000", "20": "102181884343418880000", "21": "2248001455555215360000", "22": "51704033477769953280000", "23": "1240896803466478878720000", "24": "31022420086661971968000000"}, "D": {"4": {"number": "1152", "comment": "Triality contributes the factor $6$ over the Weyl group of type $D_4$."}, "5": "3840", "6": "46080", "7": "645120", "8": "10321920", "9": "185794560", "10": "3715891200", "11": "81749606400", "12": "1961990553600", "13": "51011754393600", "14": "1428329123020800", "15": "42849873690624000", "16": "1371195958099968000", "17": "46620662575398912000", "18": "1678343852714360832000", "19": "63777066403145711616000", "20": "2551082656125828464640000", "21": "107145471557284795514880000", "22": "4714400748520531002654720000", "23": "216862434431944426122117120000", "24": "10409396852733332453861621760000"}, "D*": {"5": "3840", "6": "46080", "7": "645120", "8": "10321920", "9": "185794560", "10": "3715891200", "11": "81749606400", "12": "1961990553600", "13": "51011754393600", "14": "1428329123020800", "15": "42849873690624000", "16": "1371195958099968000", "17": "46620662575398912000", "18": "1678343852714360832000", "19": "63777066403145711616000", "20": "2551082656125828464640000", "21": "107145471557284795514880000", "22": "4714400748520531002654720000", "23": "216862434431944426122117120000", "24": "10409396852733332453861621760000"}, "E": {"6": {"number": "103680", "comment": "The full group has twice the order of the Weyl group of type $E_6$."}, "7": "2903040", "8": "696729600"}, "E*": {"6": "103680", "7": "2903040"}, "Lambda": {"9": "10321920", "10": "884736", "11": {"number": "3538944", "comment": "The catalogue lattice $\\Lambda_{11}^{\\mathrm{max}}$."}, "12": {"number": "254803968", "comment": "The catalogue lattice $\\Lambda_{12}^{\\mathrm{max}}$."}, "13": {"number": "4718592", "comment": "The catalogue lattice $\\Lambda_{13}^{\\mathrm{max}}$."}, "14": "884736", "15": "41287680", "16": {"number": "89181388800", "comment": "The Barnes-Wall lattice."}, "17": "1486356480", "18": "159252480", "19": "23592960", "20": "283115520", "21": "495452160", "22": "110361968640", "23": {"number": "84610842624000", "comment": "The order is twice that of HREF{Orders_of_sporadic_finite_simple_groups#Co2}[$\\mathrm{Co}_2$]."}, "24": {"number": "8315553613086720000", "comment": "The Leech lattice; its full group is $\\mathrm{Co}_0$, of twice the order of HREF{Orders_of_sporadic_finite_simple_groups#Co1}[$\\mathrm{Co}_1$]."}}, "K": {"12": {"number": "78382080", "comment": "The Coxeter-Todd lattice as a real lattice. Its complex-linear isometry group has index $2$, CITE{CoxeterTodd}."}}}, "Keywords": ["lattice automorphism groups", "lattice isometries", "root lattices", "laminated lattices", "Barnes-Wall lattice", "Leech lattice", "Coxeter-Todd lattice", "symmetry group orders"], "Comments": {"comment-full-group": "The group fixes the origin. No determinant restriction is imposed. It includes central inversion $-I_n$; translations are excluded. For a root lattice it contains the Weyl group and the symmetries of the Dynkin diagram. In particular, the full orders for $D_4$ and $E_6$ include factors $6$ and $2$, respectively, beyond their Weyl-group orders.", "comment-conventions": "The root lattices have minimal squared length $2$, and $\\mathbb{Z}^n$ has minimal squared length $1$. A star denotes the Euclidean dual. The catalogue matrices of $\\Lambda_n$ and $K_{12}$ have minimal squared length $4$. In dimensions $11$, $12$ and $13$, $\\Lambda_n$ denotes the catalogue variant $\\Lambda_n^{\\mathrm{max}}$, using the pages LAMBDA11, LAMBDA12 and LAMBDA13 CITE{Catalogue}. These are the same matrices as in HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices}[the classical-lattice packing table].", "comment-aliases": "The names follow HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices}[the classical-lattice packing table]: $D_3$ is listed as $A_3$; the laminated lattices in dimensions $1$ through $8$ occur under their cubic or root-lattice names; $A_1^{*}$, $A_2^{*}$ and $D_4^{*}$ are omitted because they are similar to lattices already listed; and $E_8^{*}=E_8$. The order is unchanged by scaling or duality, so a dual lattice and its root lattice have equal entries.", "comment-coxeter": "The HREF{Poincar\u00e9_polynomials_of_the_finite_Coxeter_groups}[Poincar\u00e9 polynomial] $P_W(t)=\\sum_{w\\in W}t^{\\ell(w)}$ of a Weyl group $W$, where $\\ell$ is Coxeter length, satisfies $P_W(1)=|W|$. Multiplication by the order of the Dynkin-diagram symmetry group gives the full root-lattice order.", "comment-named": "$\\Lambda_{16}$ is the Barnes-Wall lattice CITE{OEISBW}, $\\Lambda_{24}$ is the Leech lattice, and $K_{12}$ is the real Coxeter-Todd lattice CITE{CoxeterTodd}. The complex-linear isometry group of the Coxeter-Todd lattice has order $39191040$; allowing complex conjugation doubles this to the real-lattice order. The final factor $8$ printed in the real-group order on page $429$ of CITE{CoxeterTodd} is inconsistent with that doubling and with equation (16); the factor is $7$."}, "Formulas": {"formula-cubic": "$|\\operatorname{Aut}(\\mathbb{Z}^n)|=2^n n!$ for $n\\geq1$. The isometries are signed coordinate permutations.", "formula-a": "$|\\operatorname{Aut}(A_1)|=2$ and $|\\operatorname{Aut}(A_n)|=2(n+1)!$ for $n\\geq2$.", "formula-d": "$|\\operatorname{Aut}(D_4)|=1152$ and $|\\operatorname{Aut}(D_n)|=2^n n!$ for $n\\geq5$.", "formula-e": "$|\\operatorname{Aut}(E_6)|=103680$, $|\\operatorname{Aut}(E_7)|=2903040$ and $|\\operatorname{Aut}(E_8)|=696729600$.", "formula-duality": "$\\operatorname{Aut}(L)=\\operatorname{Aut}(L^{*})$ as groups of orthogonal maps on the ambient Euclidean space, and $\\operatorname{Aut}(cL)=\\operatorname{Aut}(L)$ for $c>0$. In dual bases the integral matrix correspondence is $U\\mapsto U^{-\\mathsf T}$.", "formula-sporadic": "$|\\operatorname{Aut}(\\Lambda_{23})|=2|\\mathrm{Co}_2|$ and $|\\operatorname{Aut}(\\Lambda_{24})|=2|\\mathrm{Co}_1|$, where HREF{Orders_of_sporadic_finite_simple_groups#Co2}[$\\mathrm{Co}_2$] and HREF{Orders_of_sporadic_finite_simple_groups#Co1}[$\\mathrm{Co}_1$] are Conway sporadic simple groups.", "formula-coxeter-todd": "$|\\operatorname{Aut}(K_{12})|=2^{10}3^7\\cdot5\\cdot7=78382080$, CITE{CoxeterTodd}."}, "Similar tables": [{"table": "HREF{Packing_densities_and_Hermite_numbers_of_the_classical_lattices}", "relation": "packing invariants of the same lattice families and catalogue matrices"}, {"table": "HREF{Covering_radii_and_covering_densities_of_the_classical_lattices}", "relation": "covering invariants of the same classical lattices"}, {"table": "HREF{Values_of_the_Epstein_zeta_function_of_the_classical_lattices}", "relation": "Epstein zeta values for the same lattice normalisations"}, {"table": "HREF{Poincar\u00e9_polynomials_of_the_finite_Coxeter_groups}", "relation": "evaluation at 1 gives Weyl-group orders, which differ from some full lattice-group orders"}, {"table": "HREF{Orders_of_sporadic_finite_simple_groups}", "relation": "contains the Conway-group orders related to the laminated lattices in dimensions 23 and 24"}], "References": {"CoxeterTodd": {"bib": "J. H. Conway and N. J. A. Sloane, The Coxeter-Todd lattice, the Mitchell group, and related sphere packings, Mathematical Proceedings of the Cambridge Philosophical Society 93 (1983), 421\u2013440.", "doi": "10.1017/S0305004100060746"}, "PleskenSouvignier": {"bib": "W. Plesken and B. Souvignier, Computing isometries of lattices, Journal of Symbolic Computation 24 (1997), 327\u2013334.", "doi": "10.1006/jsco.1996.0130"}}, "Programs": {"program-pari": {"code": "/* G is a positive-definite integral Gram matrix, e.g. A2. */\nG = [2,-1; -1,2];\nqfauto(G)[1] /* 12 */", "language": "PARI/GP"}}}