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❖♥ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ t✇♦ s❡ts ♦❢ ❤✐❣❤❡r✲❞✐♠❡♥s✐♦♥❛❧ ❣❛♠♠❛ ♠❛tr✐❝❡s ❛♥❞ ❛ ♣r✐♠✐t✐✈❡ ✜❡❧❞ ❡q✉❛t✐♦♥ ❉♠✐tr② ❙❤✐r♦❦♦✈ ◆❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❯♥✐✈❡rs✐t② ❍✐❣❤❡r ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s ❑❤❛r❦❡✈✐❝❤ ■♥st✐t✉t❡ ❢♦r ■♥❢♦r♠❛t✐♦♥ ❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠s ♦❢ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ❚❤❡ ✷♥❞ ❋r❡♥❝❤✲❘✉ss✐❛♥ ❝♦♥❢❡r❡♥❝❡ ✏❘❛♥❞♦♠ ●❡♦♠❡tr② ❛♥❞ P❤②s✐❝s✑ ❖❝t♦❜❡r ✶✼✲✷✶✱ ✷✵✶✻✱ P❛r✐s✱ ❋r❛♥❝❡ ❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✶ ✴ ✶✾

❉✐r❛❝ ❣❛♠♠❛ ♠❛tr✐❝❡s     ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✶ ✵ ✵ ✵ ✵ ✶ ✵     γ ✶ = γ ✵ =  ,  ,     ✵ ✵ − ✶ ✵ ✵ − ✶ ✵ ✵   − ✶ − ✶ ✵ ✵ ✵ ✵ ✵ ✵     ✵ ✵ ✵ − i ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ − ✶ i   γ ✸ =   γ ✷ =  ,  .     ✵ ✵ ✵ − ✶ ✵ ✵ ✵ i   − i ✵ ✵ ✵ ✵ ✶ ✵ ✵ γ a γ b + γ b γ a = ✷ η ab ✶ , a , b = ✵ , ✶ , ✷ , ✸ . η = � η ab � = ❞✐❛❣ ( ✶ , − ✶ , − ✶ , − ✶ ) . ❉✐r❛❝ P✳❆✳▼✳✱ Pr♦❝✳ ❘♦②✳ ❙♦❝✳ ▲♦♥❞✳ ❆✶✶✼ ✭✶✾✷✽✮✳ ❉✐r❛❝ P✳❆✳▼✳✱ Pr♦❝✳ ❘♦②✳ ❙♦❝✳ ▲♦♥❞✳ ❆✶✶✽ ✭✶✾✷✽✮✳ ❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✷ ✴ ✶✾

P❛✉❧✐✬s ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ❚❤❡♦r❡♠ ✭P❛✉❧✐✮ ❈♦♥s✐❞❡r ✷ s❡ts ♦❢ sq✉❛r❡ ❝♦♠♣❧❡① ♠❛tr✐❝❡s γ a , β a , a = ✶ , ✷ , ✸ , ✹ . ♦❢ s✐③❡ ✹ ✳ ▲❡t t❤❡s❡ ✷ s❡ts s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s γ a γ b + γ b γ a ✷ η ab ✶ , = η = ❞✐❛❣ ( ✶ , − ✶ , − ✶ , − ✶ ) , β a β b + β b β a ✷ η ab ✶ . = ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ ❝♦♠♣❧❡① ❝♦♥st❛♥t✮ ❝♦♠♣❧❡① ♠❛tr✐① T s✉❝❤ t❤❛t γ a = T − ✶ β a T , a = ✶ , ✷ , ✸ , ✹ . ❲✳P❛✉❧✐✱ ❈♦♥tr✐❜✉t✐♦♥s ♠❛t❤❡♠❛t✐q✉❡s ❛ ❧❛ t❤❡♦r✐❡ ❞❡s ♠❛tr✐❝❡s ❞❡ ❉✐r❛❝ ✱ ❆♥♥✳ ■♥st✳ ❍❡♥r✐ P♦✐♥❝❛r❡ ✻✱ ✭✶✾✸✻✮✳ ❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✸ ✴ ✶✾

❈❧✐✛♦r❞ ❛❧❣❡❜r❛s C ℓ R ( p , q ) ❛♥❞ C ℓ C ( p , q ) = C ⊗ C ℓ R ( p , q ) ❞✐♠ E = ✷ n , ❜❛s✐s✿ { e , e a ✶ , e a ✶ a ✷ , . . . , e ✶ ... n } , ❧✐♥❡❛r s♣❛❝❡ E ♦✈❡r R , ✶ ≤ a ✶ < · · · < a k ≤ n , ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ❞✐str✐❜✉t✐✈✐t②✱ ❛ss♦❝✐❛t✐✈✐t②✱ e ✲ ✐❞❡♥t✐t② ❡❧❡♠❡♥t✱ ✷ e a ✶ . . . e a k = e a ✶ ... a k , ✶ ≤ a ✶ < · · · < a k ≤ n , ✸ e a e b + e b e a = ✷ η ab e , η = || η ab || = ❞✐❛❣ ( ✶ , . . . , ✶ , − ✶ , . . . , − ✶ ) , p + q = n . � �� � � �� � p q � � u a e a + u ab e ab + · · · + u ✶ ... n e ✶ ... n = u A e A . C ℓ ( p , q ) ∋ U = ue + a a < b n � � u A e A } . C ℓ ( p , q ) = C ℓ k ( p , q ) , C ℓ k ( p , q ) = { k = ✵ | A | = k C ℓ ( p , q ) = C ℓ ❊✈❡♥ ( p , q ) ⊕ C ℓ ❖❞❞ ( p , q ) , � � C ℓ ❊✈❡♥ ( p , q ) = C ℓ k ( p , q ) , C ℓ ❖❞❞ ( p , q ) = C ℓ k ( p , q ) . k − even k − odd ❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✹ ✴ ✶✾

❚❤❡♦r❡♠ � C ℓ ✵ ( p , q ) , ✐❢ n ✲ ❡✈❡♥❀ cen C ℓ ( p , q ) = C ℓ ✵ ( p , q ) ⊕ C ℓ n ( p , q ) , ✐❢ n ✲ ♦❞❞✳ ❚❤❡♦r❡♠ ✭❈❛rt❛♥ ✶✾✵✽✱ ❇♦tt ✶✾✻✵✮  n ✷ , R ) , ▼❛t ( ✷ ✐❢ p − q ≡ ✵ ; ✷ ♠♦❞ ✽ ❀    n − ✶ n − ✶ ✷ , R ) ⊕ ▼❛t ( ✷ ✷ , R ) ,  ▼❛t ( ✷ ✐❢ p − q ≡ ✶ ♠♦❞ ✽ ❀   n − ✶ C ℓ R ( p , q ) ≃ ✷ , C ) , ▼❛t ( ✷ ✐❢ p − q ≡ ✸ ; ✼ ♠♦❞ ✽ ❀  n − ✷ ✷ , H ) ,  ▼❛t ( ✷ ✐❢ p − q ≡ ✹ ; ✻ ♠♦❞ ✽ ❀    n − ✸ n − ✸  ✷ , H ) ⊕ ▼❛t ( ✷ ✷ , H ) , ▼❛t ( ✷ ✐❢ p − q ≡ ✺ ♠♦❞ ✽ ✳ ❚❤❡♦r❡♠ � ▼❛t ( ✷ n ✷ , C ) , ✐❢ n ✲ ❡✈❡♥❀ C ℓ C ( p , q ) ≃ n − ✶ n − ✶ ✷ , C ) ⊕ ▼❛t ( ✷ ✷ , C ) , ▼❛t ( ✷ ✐❢ n ✲ ♦❞❞✳ ❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✺ ✴ ✶✾

▲❡t t❤❡ s❡t ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ❡❧❡♠❡♥ts s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s β a ∈ C ℓ ( p , q ) , β a β b + β b β a = ✷ η ab e . ❚❤❡♥ t❤❡ s❡t γ a = T − ✶ β a T , ∀ ✐♥✈❡rt✐❜❧❡ T ∈ C ℓ ( p , q ) s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s γ a γ b + γ b γ a = ✷ η ab e . ❘❡❛❧❧②✱ γ a γ b + γ b γ a = T − ✶ β a TT − ✶ β b T + T − ✶ β b TT − ✶ β a T = = T − ✶ ( β a β b + β b β a ) T = T − ✶ ✷ η ab eT = ✷ η ab e . ❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✻ ✴ ✶✾

❚❤❡♦r❡♠ ✭❈❛s❡ ♦❢ ❡✈❡♥ n ✮ ❈♦♥s✐❞❡r r❡❛❧ ✭♦r ❝♦♠♣❧❡①✐✜❡❞✮ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ C ℓ ( p , q ) ♦❢ ❡✈❡♥ ❞✐♠❡♥s✐♦♥ n = p + q ✳ ▲❡t t❤❡ ❢♦❧❧♦✇✐♥❣ ✷ s❡ts ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ❡❧❡♠❡♥ts γ a , β a , a = ✶ , ✷ , . . . , n s❛t✐s❢② ❝♦♥❞✐t✐♦♥s γ a γ b + γ b γ a = ✷ η ab e , β a β b + β b β a = ✷ η ab e . ❚❤❡♥ ❜♦t❤ s❡ts ♦❢ ❡❧❡♠❡♥ts ❣❡♥❡r❛t❡ ❜❛s❡s ♦❢ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ r❡❛❧ ✭❝♦♠♣❧❡①✮ ❝♦♥st❛♥t✮ ❡❧❡♠❡♥t T ∈ C ℓ ( p , q ) s✉❝❤ t❤❛t γ a = T − ✶ β a T , ∀ a = ✶ , . . . , n . ▼♦r❡♦✈❡r✱ ✇❡ ❝❛♥ ❛❧✇❛②s ✜♥❞ t❤✐s ❡❧❡♠❡♥t T ✐♥ t❤❡ ❢♦r♠ � γ A = ( γ A ) − ✶ β A F γ A , T = A ✇❤❡r❡ F ✐s ❛♥② ❡❧❡♠❡♥t ♦❢ ❛ s❡t ✐❢ β ✶ ... n � = − γ ✶ ... n ; ✐❢ β ✶ ... n � = γ ✶ ... n ✶ ) { γ A , A ∈ I ❊✈❡♥ } ✷ ) { γ A , A ∈ I ❖❞❞ } s✉❝❤ t❤❛t ❝♦rr❡s♣♦♥❞✐♥❣ T ✐s ♥♦♥③❡r♦ T � = ✵ ✳ ❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✼ ✴ ✶✾

❈❛s❡ ♦❢ ❡✈❡♥ n ✐♥ ♠❛tr✐① ❢♦r♠❛❧✐s♠ ❚❤❡♦r❡♠ ▲❡t n ✲ ❡✈❡♥ ❛♥❞ ✷ s❡ts ♦❢ sq✉❛r❡ ♠❛tr✐❝❡s γ a , β a , a = ✶ , ✷ , . . . , n s❛t✐s❢② ❝♦♥❞✐t✐♦♥s γ a γ b + γ b γ a = ✷ η ab ✶ , β a β b + β b β a = ✷ η ab ✶ . n ✷ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ❛ ■❢ ♠❛tr✐❝❡s ❛r❡ ❝♦♠♣❧❡① ♦❢ t❤❡ ♦r❞❡r ✷ ❝♦♠♣❧❡① ❝♦♥st❛♥t✮ ♠❛tr✐① T s✉❝❤ t❤❛t n ✷ ✱ t❤❡♥ ■❢ s✐❣♥❛t✉r❡ ✐s p − q ≡ ✵ , ✷ ♠♦❞ ✽ ❛♥❞ ♠❛tr✐❝❡s ❛r❡ r❡❛❧ ♦❢ t❤❡ ♦r❞❡r ✷ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ❛ r❡❛❧ ❝♦♥st❛♥t✮ ♠❛tr✐① T s✉❝❤ t❤❛t ■❢ s✐❣♥❛t✉r❡ ✐s p − q ≡ ✹ , ✻ ♠♦❞ ✽ ❛♥❞ ♠❛tr✐❝❡s ❛r❡ ♦✈❡r t❤❡ q✉❛t❡r♥✐♦♥s ♦❢ n − ✷ ✷ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ❛ r❡❛❧ ❝♦♥st❛♥t✮ ♠❛tr✐① T t❤❡ ♦r❞❡r ✷ s✉❝❤ t❤❛t γ a = T − ✶ β a T , a = ✶ , . . . n . ❉♠✐tr② ❙❤✐r♦❦♦✈ P❛r✐s✱ ❋r❛♥❝❡ ✷✵✶✻ ✽ ✴ ✶✾