By M. Aizenman (Chief Editor)

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**Extra resources for Communications in Mathematical Physics - Volume 245**

**Example text**

LS ) , then the dual tensor (λ ,... ,λ ) S αI ∈ (S)1 (M) is in the irreducible representation of O(D − 1, 1) associated to I . the Young diagram Y(λ 1 ,... 9). 9) for all I only in the exceptional case where D is even and li = lj = D/2. Note that this condition is satisfied for free gauge theories of completely symmetric tensors in D = 4 flat space. 8) is generally not satisfied by the dual tensor KI . 8) is not satisfied. A natural idea is that when li + lj > D for a curvature tensor (l1 ,...

E. (S) (S) Y(n1 ,... ,nS ) ⇔ mi ≤ ni ≤ mi + 1 Y(m1 ,... ,mS ) ∀i ∈ {1, . . , S} . z< . zz z zz zz (3,3) / ~·? · · > •O ~ } ~ } ~~ }} ~~ }} ~ } } ~ (2,2) / (3,2) / ·? · · ~~ }> •O }> •O ~ } } ~~ }} }} ~~ }} }} ~ } } } } ~ (1,1) / (2,1) / (3,1) / ·? 6) where all the arrows represent maps . This diagram is completely commutative. The previous inclusions ⊂ and provide partial order relations for Y(S) . The order is only partial because all Young tableaux are not comparable. We now introduce a total order relation for Y(S) .

I=1 Order relations. For future reference, the subset Y(S) of NS is defined by Y(S) ≡ {(n1 , . . , nS ) ∈ NS | n1 ≥ n2 ≥ ... ≥ nS ≥ 0} . 2) For two columns, the set Y(2) can be depicted as the following set of points in the plane R2 : ... (3,3) • (2,2) (3,2) • • (1,1) (2,1) (3,1) • • • (0,0) (1,0) (2,0) (3,0) • • • • ... 3) ... , lS . If Yp is a welldefined Young diagram, then (l1 , l2 , . . , lS ) ∈ Y(S) . Conversely, a Young diagram Y with S columns is uniquely determined by the gift of an element of Y(S) , and can there(S) fore be labeled unambiguously as Y(l1 ,l2 ,...

### Communications in Mathematical Physics - Volume 245 by M. Aizenman (Chief Editor)

by Kenneth

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