The checkerboard lattice and the type Dₙ discriminant form #
This file constructs the checkerboard lattice {x ∈ ℤⁿ | ∑ xᵢ is even} inside ℚⁿ, computes
its dual lattice, and identifies its discriminant form. For n ≥ 4 this is the positive root
lattice of type Dₙ in the Conway--Sloane coordinate model, and the discriminant form computed
here is the Dₙ row of the ADE table. The same coordinate formula defines the underlying lattice
at every rank. The dual description, named representatives, and order-four discriminant results
below assume n > 0; they include the low ranks 1 ≤ n ≤ 3, at which the Dynkin type Dₙ is not
defined and where the lattice may have no roots at all, as for n = 1. None of those results needs
4 ≤ n, so the declarations are named after the coordinate model rather than after the Dynkin type.
The lattice carries the standard dot product. It is even, because a sum of squares of integers is
congruent modulo two to the sum of those integers. Its dual lattice is ℤⁿ ∪ (s + ℤⁿ), described
here by the two conditions that all coordinate differences are integers and that the doubled last
coordinate is an integer.
The discriminant group is generated by the three Conway--Sloane representatives
- the vector class
v = eₙ, of half-norm1 / 2; - the spinor class
s = (e₁ + ⋯ + eₙ) / 2, of half-normn / 8; - the cospinor class
c = s - eₙ, again of half-normn / 8.
These three classes together with the zero class exhaust the discriminant group, which therefore
has order four; the form is positive definite, so the Gram determinant in any integral basis is
4. The group is cyclic of order four when n is odd, generated by the spinor class with
2s = v, and is (ℤ/2)² when n is even. The two spinor classes then carry the same quadratic
value, and their mutual pairing is b(s, c) = (n - 2) / 4.
The file goes further and identifies the discriminant form itself, not only its group. For even
n the quadratic values q(v) = 1 / 2 and q(s) = q(c) = n / 8 together with the pairing
b(v, s) = 1 / 2 present it on (ℤ/2)², and checkerboardDiscriminantQuadraticIsometry is an
isometry of finite quadratic modules from that presented model onto A_L. Group order alone does
not determine this form: as n runs over the even residues modulo eight the model runs through
Nikulin's u₁ (for n ≡ 0), his v₁ (for n ≡ 4), and the two forms with quarter-integral
spinor values (for n ≡ 2 mod 4).
For odd n the single value q(s) = n / 8 already presents the form, because s generates the
cyclic group ℤ/4: checkerboardCyclicQuadraticModule is the cyclic model and
checkerboardCyclicQuadraticIsometry is an isometry from it onto A_L. The model records the
whole of the odd Dₙ table row, since its value at 2 is the vector value q(v) = 1 / 2 — an
equality which holds in ℚ/ℤ exactly because n is odd, and 2 is carried to v by
checkerboardCyclicQuadraticIsometry_two — and its generator self-pairing is b(s, s) = n / 4.
The representatives are the ones fixed by Conway and Sloane, so that later glue calculations —
in particular the enlargement of D₈ to E₈ — can reuse them without a change of
representative. The identification of this coordinate model with the root lattice of type
Dₙ — a ℤ-basis of Bourbaki simple roots whose Gram matrix is CartanMatrix.D n — is in
TauCeti.LinearAlgebra.IntegralLattice.RootLattice.TypeD.SimpleRoots.
Main definitions #
TauCeti.IntegralLattice.checkerboardCarrier: the integer vectors ofℚⁿwith even coordinate sum.TauCeti.IntegralLattice.checkerboardLattice: the checkerboard lattice itself, the typeDₙroot lattice forn ≥ 4.TauCeti.IntegralLattice.checkerboardVector,checkerboardSpinor,checkerboardCospinor: the Conway--Sloane representativesv,s,cof the nonzero discriminant classes.TauCeti.IntegralLattice.checkerboardVectorClass,checkerboardSpinorClass,checkerboardCospinorClass: their classes in the discriminant group.TauCeti.IntegralLattice.zmodFourAddEquivCheckerboardDiscriminantGroup: the discriminant group of an odd-rank checkerboard lattice isℤ/4.TauCeti.IntegralLattice.zmodTwoProdAddEquivCheckerboardDiscriminantGroup: the discriminant group of an even-rank checkerboard lattice is(ℤ/2)².TauCeti.IntegralLattice.checkerboardStandardQuadraticModule: the presented(ℤ/2)²model of the discriminant form of an even-rank checkerboard lattice.TauCeti.IntegralLattice.checkerboardCyclicQuadraticModule: the presentedℤ/4model of the discriminant form of an odd-rank checkerboard lattice.
Main results #
TauCeti.IntegralLattice.isEven_checkerboardLattice: the checkerboard lattice is even.TauCeti.IntegralLattice.isPosDef_checkerboardLattice: it is positive definite.TauCeti.IntegralLattice.mem_checkerboardLattice_dualCarrier_iff: the dual lattice.checkerboardDiscriminantGroup_eq_zero_or_vectorClass_or_spinorClass_or_cospinorClass: the four classes exhaust the discriminant group.TauCeti.IntegralLattice.natCard_discriminantGroup_checkerboardLatticeanddiscriminant_checkerboardLattice: the discriminant group has order four.TauCeti.IntegralLattice.discriminantQuadraticMap_checkerboardVectorClass,discriminantQuadraticMap_checkerboardSpinorClass,discriminantQuadraticMap_checkerboardCospinorClass: the quadratic values1 / 2,n / 8,n / 8.TauCeti.IntegralLattice.discriminantPairing_checkerboardSpinorClass_checkerboardCospinorClass: the mutual pairing(n - 2) / 4of the spinor and cospinor classes.TauCeti.IntegralLattice.discriminantPairing_checkerboardVectorClass_checkerboardSpinorClass,discriminantPairing_checkerboardVectorClass_self,discriminantPairing_checkerboardSpinorClass_self: the remaining pairing values1 / 2,0andn / 4.TauCeti.IntegralLattice.checkerboardDiscriminantQuadraticIsometry: for evenn, the presented model is isometric to the discriminant quadratic module.TauCeti.IntegralLattice.checkerboardCyclicQuadraticIsometry: for oddn, the cyclic model is isometric to the discriminant quadratic module.TauCeti.IntegralLattice.isNondegenerate_checkerboardStandardQuadraticModuleandisNondegenerate_checkerboardCyclicQuadraticModule: the presented models are nondegenerate.
References #
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, §4.7.1.
- V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, §1.
- W. Ebeling, Lattices and Codes, Chapter 1.
TauCetiRoadmap/IntegralLattices/README.md, Layer 5, theDₙrows of the ADE table.
The checkerboard carrier is a full lattice in ℚⁿ.
The lattice #
The checkerboard lattice in the Conway–Sloane coordinate model: the integer vectors of
ℚⁿ whose coordinate sum is even, with the standard dot product. For n ≥ 4 this is the
positive root lattice of type Dₙ.
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The carrier of the checkerboard lattice.
The rational form of the checkerboard lattice is the identity Gram form.
The form of the checkerboard lattice is the standard dot product.
The checkerboard lattice Dₙ is positive definite: its form is the standard dot product
of ℚⁿ.
Membership in the checkerboard lattice: every coordinate is an integer and their sum is even.
The checkerboard lattice is nondegenerate, the identity Gram matrix being invertible.
The checkerboard lattice is even: the sum of squares of integers with even sum is itself even.
The dual lattice #
The final coordinate index n - 1 of Fin n. The Conway–Sloane vector class of the
checkerboard discriminant group is the standard vector at this index.
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The underlying natural number of the final coordinate index is n - 1.
The dual of the checkerboard lattice: a rational vector pairs integrally with every
lattice vector exactly when all of its coordinate differences are integers and its doubled last
coordinate is an integer. Equivalently the dual lattice is ℤⁿ ∪ (s + ℤⁿ) for the
half-sum s.
The three nontrivial discriminant classes #
The Conway–Sloane vector representative v = eₙ of the checkerboard discriminant group.
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The Conway–Sloane spinor representative s = (e₁ + ⋯ + eₙ) / 2.
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- TauCeti.IntegralLattice.checkerboardSpinor n x✝ = 1 / 2
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The Conway–Sloane cospinor representative c = s - eₙ.
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The coordinates of the vector representative.
The coordinates of the spinor representative.
The coordinates of the cospinor representative.
The vector representative lies in the dual lattice.
The spinor representative lies in the dual lattice.
The cospinor representative lies in the dual lattice.
Pairings among the representatives #
The last coordinate of the vector representative is 1.
The last coordinate of the cospinor representative is -1 / 2.
The coordinate sum of the vector representative is 1.
The coordinate sum of the spinor representative is n / 2.
The coordinate sum of the cospinor representative is n / 2 - 1.
Pairing an arbitrary vector against the vector representative reads off its last coordinate.
Pairing an arbitrary vector against the spinor representative halves its coordinate sum.
Pairing an arbitrary vector against the cospinor representative.
The vector representative has norm one.
The spinor representative has norm n / 4.
The cospinor representative has norm n / 4.
The vector and spinor representatives pair to 1 / 2.
The spinor and cospinor representatives pair to (n - 2) / 4.
The discriminant group #
The class of the vector representative v in the checkerboard discriminant group.
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The class of the spinor representative s in the checkerboard discriminant group.
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The spinor class is represented by the Conway--Sloane spinor vector.
The class of the cospinor representative c in the checkerboard discriminant group.
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Two dual vectors define the same discriminant class as soon as their difference is an integer vector with even coordinate sum.
The four Conway–Sloane classes exhaust the checkerboard discriminant group.
The four classes are distinct #
The vector representative is not a lattice vector: its coordinate sum is odd.
The spinor representative is not a lattice vector: its coordinates are half-integral.
The cospinor representative is not a lattice vector.
The cospinor and vector representatives differ by a half-integral vector.
The spinor and cospinor representatives differ by a vector of odd coordinate sum.
The vector class is nonzero.
The spinor class is nonzero.
The cospinor class is nonzero.
The spinor class differs from the vector class.
The cospinor class differs from the vector class.
The spinor class differs from the cospinor class.
The order of the discriminant group #
The checkerboard discriminant group has order four.
The discriminant of the checkerboard lattice is 4. The form is positive definite, so
the Gram determinant of any integral basis is 4 itself.
The discriminant quadratic form on the three classes #
The vector class has quadratic value 1 / 2.
The spinor class has quadratic value n / 8.
The cospinor class also has quadratic value n / 8.
The spinor and cospinor classes pair to (n - 2) / 4.
The group structure #
The cospinor class is the difference of the spinor and vector classes.
The vector class has order two: 2 eₙ is a lattice vector for every n.
For even n the spinor class has order two: the all-ones vector 2 s has even
coordinate sum.
For even n the spinor class has additive order two.
For even n, the multiples of the spinor class are exactly zero and the spinor class.
For odd n the vector class is twice the spinor class, so the discriminant group is
cyclic of order four.
The sum of the vector and spinor classes is the cospinor class.
For odd n the checkerboard discriminant group is cyclic of order four, generated by the
spinor class.
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The odd-rank identification sends the generator 1 of ℤ/4 to the spinor class.
For even n the checkerboard discriminant group is (ℤ/2)², with the two factors
generated by the vector class and by the spinor class.
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The even-rank identification sends the first generator of (ℤ/2)² to the vector class.
The even-rank identification sends the second generator of (ℤ/2)² to the spinor class.
The even-rank identification sends the diagonal generator of (ℤ/2)² to the cospinor
class.
The discriminant pairings #
The vector class is bilinear-isotropic: the ambient self-pairing ⟨v, v⟩ = 1 is an
integer, so b(v, v) = 0.
The spinor class has self-pairing b(s, s) = n / 4.
The vector and spinor classes pair to 1 / 2.
The discriminant quadratic module of an even-rank checkerboard lattice #
The standard (ℤ/2)² model of the even-rank checkerboard discriminant form: the vector
class has quadratic value 1 / 2, either spinor class has n / 8, and the vector and spinor
classes pair to 1 / 2.
For n ≡ 0 mod 8 this is Nikulin's u₁, for n ≡ 4 mod 8 it is v₁, and for n ≡ 2 mod 4 the
two spinor classes carry the quarter-integral values n / 8.
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The first standard generator has quadratic value 1 / 2.
The standard model is isometric to the discriminant quadratic module of an even-rank
checkerboard lattice, by the identification carrying (1, 0) to the vector class and (0, 1)
to the spinor class.
This is the Dₙ row of the ADE table for even n: not only is the discriminant group (ℤ/2)²,
its quadratic form is the displayed one.
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The standard (ℤ/2)² model is nondegenerate, since the discriminant form of a
nondegenerate lattice is.
The discriminant quadratic module of an odd-rank checkerboard lattice #
The cyclic ℤ/4 model of the odd-rank checkerboard discriminant form: the generator has
quadratic value n / 8.
The two torsion conditions demanded by the cyclic construction are 16 · (n / 8) = 2n and
8 · (n / 8) = n, both integers, so the model is defined for every n; it is the discriminant
form of the checkerboard lattice exactly when n is odd, since only then does the spinor class
generate.
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The generator of the cyclic model has quadratic value n / 8.
The double of the generator carries the vector value 1 / 2, for odd n. The underlying
computation is 4 · (n / 8) = n / 2, which is 1 / 2 in ℚ/ℤ exactly when n is odd.
The generator of the cyclic model has self-pairing n / 4.
The cyclic model is isometric to the discriminant quadratic module of an odd-rank
checkerboard lattice, by the identification carrying 1 to the spinor class.
This is the Dₙ row of the ADE table for odd n: not only is the discriminant group ℤ/4, its
quadratic form is the displayed one. A single generator value suffices here, in contrast to the
three values needed in the even-rank case.
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The odd-rank quadratic isometry acts through the discriminant-group equivalence.
The odd-rank quadratic isometry carries the generator of ℤ/4 to the spinor class.
The odd-rank quadratic isometry carries the element 2 of ℤ/4 to the vector class.
Together with checkerboardCyclicQuadraticModule_quadratic_two this reads the vector value
q(v) = 1 / 2 of the table row off the model.
The cyclic ℤ/4 model is nondegenerate, since the discriminant form of a nondegenerate
lattice is.