# dga6 - TENSOR PRODUCTS OF SYMMETRIC FUNCTIONS OVER Z 2 KARL...

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Unformatted text preview: TENSOR PRODUCTS OF SYMMETRIC FUNCTIONS OVER Z 2 KARL HEINZ DOVERMANN AND JASON HANSON Abstract. We calculate the homology and the cycles in tensor prod- ucts of algebras of symmetric function over Z 2 . 1. Statement of Results We calculate the homology and the cycles in tensor products of the differ- ential graded algebra of symmetric functions over Z 2 , the integers modulo 2. The need for this calculation arises in a project in which we show that closed smooth manifolds with cyclic group actions have equivariant real algebraic models ([3] and [4]). There we need to calculate the ordinary equivariant co- homology of some classifying spaces with cyclic group action; the symmetric functions and the differential ∇ arise in a way explained in some detail in Section 5. Some special cases of our results can be extracted from [6] and [5]. Let F a := Z 2 [ z 1 , . . . , z a ] be the polynomial ring in a variables of dimen- sion 1. The natural differential ∇ on F a is the sum of the partial derivatives. This derivative is obtained from the standard rules of differentiation: lin- earlity and Leibnitz’ rule (the product rule), under the assumption that the derivative of each z i is the constant function 1. Then ∇ 2 = 0, and ( F a , ∇ ) is a differential graded algebra. In F a we consider the subalgebra S a of symmetric functions. The i –th elementary symmetric function is denoted by σ i . It is elementary to compute its derivative: (1.1) ∇ σ i = ( a- i + 1) σ i- 1 for 1 ≤ i ≤ a , and ∇ σ = 0 . The formula depends only on the parity of a . Consider a sequence A = ( a (0) , . . . , a ( k )) of nonnegative integers and set (1.2) S A = S a (0) ⊗ ··· ⊗ S a ( k ) . As a tensor product, S A inherits a natural differential operator, which we still denote by ∇ . We use an additional subscript to distinguish the factor to which an elementary symmetric function belongs. Specifically, σ j,i is the Date : February 26, 2006. 1991 Mathematics Subject Classification. Primary 13D07, 13N10. Key words and phrases. Symmetric Functions, Differential Graded Algebras. 1 2 KARL HEINZ DOVERMANN AND JASON HANSON i –th elementary symmetric function in the j –th factor of the tensor product, where 0 ≤ j ≤ k and 0 ≤ i ≤ a ( j ). We set a ( j ) = 2 b ( j ) or a ( j ) = 2 b ( j ) + 1, depending on whether a ( j ) is even or odd. The letter Z denotes the cycles of the indicated DGA. Theorem 1.1. Suppose A = ( a (0) , . . . , a ( k )) and for a distinguished index t the associated entry a ( t ) of A is odd. Then the differential graded algebra S A is acyclic. Set D ( A ) = { σ t, 1 + σ j, 1 | j 6 = t, a ( j ) odd } D o ( A ) = { σ j, 2 s , σ t, 1 σ j, 2 s + σ j, 2 s +1 | a ( j ) = 2 b ( j ) + 1 and 1 ≤ s ≤ b ( j ) } D e ( A ) = { σ j, 2 s- 1 , σ t, 1 σ j, 2 s- 1 + σ j, 2 s | a ( j ) = 2 b ( j ) and 1 ≤ s ≤ b ( j ) } ....
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dga6 - TENSOR PRODUCTS OF SYMMETRIC FUNCTIONS OVER Z 2 KARL...

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