Proposition. Exterior weights give the finite spin characters [fcap-0030]
✍️source

Work over the complex split models and positive systems fixed in the preceding type-B and type-D cards. In type \(B_\ell \) assume \(\ell \geq 1\); in type \(D_\ell \) assume \(\ell \geq 4\). For \(I\subseteq \{1,\ldots ,\ell \}\), put \[\lambda _I=\tfrac 12\left ( \sum _{i\in I}\epsilon _i-\sum _{i\notin I}\epsilon _i\right ).\] By Theorem [fcap-0025], the line \(\mathbb C u_I\) has weight \(\lambda _I\); distinct subsets give distinct weights. Consequently \[\operatorname {ch}S_{B_\ell } =\sum _I e^{\lambda _I} =\prod _{i=1}^{\ell } \left (e^{\epsilon _i/2}+e^{-\epsilon _i/2}\right ),\] and the type-D parity carriers have \[\operatorname {ch}S_D^+ =\sum _{\lvert I\rvert \ {\rm even}}e^{\lambda _I},\qquad \operatorname {ch}S_D^- =\sum _{\lvert I\rvert \ {\rm odd}}e^{\lambda _I}.\] If \(a_i=e^{\epsilon _i/2}\) and \(b_i=e^{-\epsilon _i/2}\), sorting the product by occupied parity gives \[\operatorname {ch}S_D^\pm =\tfrac 12\left [ \prod _i(a_i+b_i)\ \mathbin {\pm }\ \prod _i(b_i-a_i)\right ].\] The order \(b_i-a_i\) makes an occupied coordinate contribute the minus sign. These are finite sums obtained from the exterior basis, without invoking a general character formula.

The complete half-sum weights above become honest characters on the Spin torus. Put \(A_i=e_i e_i'\), so its action is \(N_i\). For \(t\in \mathbb C^\times \), define \[g_i(t)=tA_i+t^{-1}(1-A_i) =(t e_i+t^{-1}e_i')(e_i+e_i').\] Both vector factors have quadratic value \(1\); the pair criterion therefore puts \(g_i(t)\) in the actual Spin carrier CliffordAlgebra.ι_mul_ι_mem_spinGroup_of_norm_mul_norm_eq_one. The elements for different coordinates commute, and \[g_i(t)u_I= \begin {cases}t\,u_I,&i\in I,\\t^{-1}u_I,&i\notin I.\end {cases}\] Their vector action scales \(e_i\) by \(t^2\), \(e_i'\) by \(t^{-2}\), and fixes the other coordinates. Thus \(g(t_1,\ldots ,t_\ell )=\prod _i g_i(t_i)\) covers the standard orthogonal torus by coordinatewise squaring.

The kernel of this parameterization is \[\{(t_i):t_i\in \{1,-1\},\ \prod _i t_i=1\}.\] Indeed, equality of its spinor action with the identity forces \(t_i^2=1\) by comparing weights that differ in coordinate \(i\), and the vacuum weight then forces \(\prod _i t_i=1\). Conversely such a tuple gives the actual Clifford scalar \(\prod _i t_i=1\). Thus both implications hold directly for the group parameterization. The image contains the scalar \(-1\); together with the usual \(\{1,-1\}\) kernel of \(\operatorname {Spin}\to SO\) [fulton2004representation, Proposition 20.28, pp. 308--309], this is the full inverse-image torus.

A Laurent monomial on the parameter torus descends through the displayed sign kernel exactly when all its integer exponents have the same parity. Relative to the orthogonal coordinates \(x_i=t_i^2\), the Spin weight lattice is therefore \[P=\mathbb Z^\ell \cup \left (\mathbb Z+\tfrac 12\right )^\ell .\] The complete signed sums \(\lambda _I\) lie in \(P\), while an individual symbol \(e^{\epsilon _i/2}\) need not itself be a character. The orthogonal torus has character lattice \(\mathbb Z^\ell \); every spinor half-sum takes the scalar \(-1\) to \(-1\) and hence does not descend to it. In type D the adjoint torus has the still smaller root lattice.

[goodman2009symmetry, Corollary 6.3.9, p. 322; Sections 7.1 and 7.2.3, pp. 330 and 346] independently identify complex Spin as algebraically simply connected, explain the single-valued square-root torus factors, and identify the torus-character lattice with the weight lattice. Those group results support the character interpretation. The pinned TauCeti results used above are Lie and Clifford statements; no formal integration or differentiation comparison is asserted.

For \(B_2\), the four distinct weights are summarized by \[\operatorname {ch}S_{B_2} =(x^{1/2}+x^{-1/2})(y^{1/2}+y^{-1/2}),\] with highest weight \((\epsilon _1+\epsilon _2)/2=\omega _2\). For \(D_4\), the even- and odd-cardinality sums contain eight weights each: \(\omega _4=(\epsilon _1+\epsilon _2+\epsilon _3+\epsilon _4)/2\) lies in \(S^+\), while \(\omega _3=(\epsilon _1+\epsilon _2+\epsilon _3-\epsilon _4)/2\) lies in \(S^-\). This is a finite character check, not a triality claim.

Proposition. Exterior weights give the finite spin characters [fcap-0030]
✍️source

Reference. Symmetry, Representations, and Invariants [goodman2009symmetry]

Reference. Representation theory: A first course [fulton2004representation]

Polarization and the exterior spinor model [fcap-001G]

Definition. Polarization data and its carrier [chevalley1954algebraic, II.2.1--II.2.2, pp. 42--44] [fcap-001H]

SpinPolarizationData
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Convention. Clifford generators and polarized operators [fcap-0027]

Definition. The three polarized Clifford operators [fcap-001I]

wedge contract lineOperator cliffordOperator
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Lemma. The polarized Clifford relation [meinrenken2013clifford, Section 3.2.1, pp. 54--55] [fcap-001J]

contract_wedge cliffordOperator_sq
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Theorem. The exterior Clifford action [meinrenken2013clifford, Section 3.2.2, Proposition 3.5, p. 57] [fcap-001K]

spinAction
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Theorem. Creation and contraction generate all endomorphisms [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60] [fcap-001L]

creation_contraction_adjoin_eq_top spinAction_surjective
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Definition. The Spin and Pin representations [meinrenken2013clifford, Section 3.2.1, pp. 55--56] [fcap-001M]

spinRep pinRep
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Lemma. The restricted actions are the Clifford action [meinrenken2013clifford, Section 3.2.1, pp. 55--56] [fcap-001N]

spinRep_apply pinRep_apply
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Example. Rank one: matrix units and occupation [fcap-001P]

Theorem. Matrix units give the even-dimensional structure theorem [fcap-002V]

Proposition. An odd spinor action selects one Clifford summand [fcap-002W]

Theorem. Exterior pairings determine compact spinor reality [fcap-0041]

Proposition. Real matrix blocks control Spin module complexification [fcap-0042]

Theorem. Occupation signs split the half-spin carrier [fcap-0025]

spinAction_diagonalBivector_basis iSup_spinWeightSpace_even_eq_spinPlus iSup_spinWeightSpace_odd_eq_spinMinus
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Proposition. The simple type-D operators generate the positive nilradical [fcap-002X]

Theorem. The two top exterior words are the type-D fork highest vectors [fcap-002Y]

Theorem. The top exterior word is the type-B spin highest vector [fcap-002Z]

Proposition. Exterior weights give the finite spin characters [fcap-0030]
✍️source

Proposition. Complex Spin in dimension three is \(SL_2\) on the spin module [fcap-0031]

Proposition. Complex Spin in dimension four is two copies of \(SL_2\) [fcap-0032]

Proposition. Complex Spin in dimension five is the symplectic group on spinors [fcap-0033]

Proposition. Complex Spin in dimension six is \(SL_4\) on a half-spin module [fcap-0034]

Proposition. The outer nodes of \(D_4\) permute the three eight-dimensional classes [fcap-003B]

Theorem. The actual compact \(\operatorname {Spin}(8)\) has outer Lie-automorphism group \(S_3\) [fcap-003C]

Proposition. Clifford multiplication gives the triality trilinear [fcap-003D]

Theorem. A pinned three-cycle integrates to compact triality [fcap-003T]

Proof.

Theorem. Triality cycles three real modules and preserves the trilinear [fcap-003U]

Proof.

Remark. Horizons beyond the exterior model [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60]; [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287]; [lawson2016spin, I.2, pp. 13--19] [fcap-0017]

Remark. Horizons beyond the exterior model [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60]; [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287]; [lawson2016spin, I.2, pp. 13--19] [fcap-0017]

Proposition. The outer nodes of \(D_4\) permute the three eight-dimensional classes [fcap-003B]

Theorem. Occupation signs split the half-spin carrier [fcap-0025]

spinAction_diagonalBivector_basis iSup_spinWeightSpace_even_eq_spinPlus iSup_spinWeightSpace_odd_eq_spinMinus
L∃∀N