Hochschild cohomology of symmetric groups and generating functions

The generating functions for the dimensions of the Hochschild cohomology of the group algebra of the symmetric group S n on 𝑛 letters in low degrees are given by

For any integer r ⩾ 0 , there exists a rational function R p , r ⁢ ( t ) with integer coefficients such that

Let 𝑛, 𝑘 be integers such that n ⩾ 0 and k ⩾ 1 . Define

Thus F k ⁢ ( n ) is the total number of parts of length 𝑘 in all partitions of 𝑛.

Let 𝑛, 𝑘 be integers such that n ⩾ k ⩾ 1 . We have

Proof

Removing a part of length 𝑘 gives a bijection between the partitions of 𝑛 with a part of length 𝑘 and the partitions of n − k , which reduces by one the number of parts of length 𝑘. ∎

Let 𝑛, 𝑘 be integers such that n ⩾ 0 and k ⩾ 1 . We have

In particular, F k ⁢ ( n ) = 0 for n < k . The generating function for F k ⁢ ( n ) is given by

Proof

The first part follows by induction from Lemma 2.2. Thus

Let 𝑛 be a non-negative integer. If λ ⊢ n and 𝑘, ℓ are positive integers, let g k , ℓ ⁢ ( λ ) be the number of distinct lengths divisible by 𝑘 of parts of 𝜆 that are repeated at least ℓ times. More precisely, if λ = ( n λ n ⁢ … ⁢ 2 λ 2 ⁢ 1 λ 1 ) , then g k , ℓ ⁢ ( λ ) is the number of indices 𝑖, 1 ⩽ i ⩽ n , such that 𝑘 divides 𝑖 and λ i ⩾ ℓ . Let G k , ℓ ⁢ ( n ) = ∑ λ ⊢ n g k , ℓ ⁢ ( λ ) .

For 𝑛 a non-negative integer and 𝑘, ℓ positive integers, we have

Proof

Consider the set G k , ℓ ⁢ ( n ) of ordered pairs ( λ , i ) , where λ ⊢ n is a partition containing a part of length i ⁢ k repeated at least ℓ times. Then G k , ℓ ⁢ ( n ) = | G k , ℓ ⁢ ( n ) | . We have a bijection from G k , ℓ ⁢ ( n ) to

given by removing ℓ parts of length i ⁢ k from 𝜆. ∎

For 𝑛 a non-negative integer and 𝑘, ℓ positive integers, we have

Proof

This follows from Proposition 2.3 and Lemma 2.5. ∎

This identity with k = 1 can be found in Stanley’s book [10, Exercise 80 in Chapter 1]. This is commonly known as Elder’s theorem, and the case k = 1 , ℓ = 1 is known as Stanley’s theorem. In the solution to the exercise in [10], Stanley explains that he discovered the result for k = 1 and all ℓ ⩾ 1 in 1972 and submitted it to the “Problems and Solutions” section of Amer. Math. Monthly, where it was rejected with the comment “A bit on the easy side, and using only a standard argument.” The statement was reproved by Elder in 1984, and other proofs can be found in Hoare [7] and Kirdar and Skyrme [8]. But as pointed out by Gilbert [5], all these proofs were predated by more than a decade in a 1959 paper of Fine [4]. A different generalisation of Elder’s theorem was proved by Andrews and Deutsch [1].

For a non-negative integer 𝑛 and a positive integer 𝑟, let q ⁢ ( n , r ) be the number of partitions of 𝑛 into 𝑟 distinct parts.

Let 𝑟 be a positive integer. The generating function for q ⁢ ( n , r ) , n ⩾ 0 , is given by

Proof

By going over to conjugate partitions, one sees that q ⁢ ( n , r ) is also equal to the number of partitions of 𝑛 with each of 1 , 2 , … , r occurring as a part and with 𝑟 as largest part. Thus the generating function equals

Let 𝑛 be a non-negative integer. If λ ⊢ n and 𝑘, 𝑟 are positive integers, let c k , r ⁢ ( λ ) be the number of unordered 𝑟-tuples of distinct part lengths of 𝜆 divisible by 𝑘. Let C k , r ⁢ ( n ) = ∑ λ ⊢ n c k , r ⁢ ( λ ) and let D k , r ⁢ ( n ) = ∑ λ ⊢ n ( g 1 , k ⁢ ( λ ) r ) .

Let 𝑛 be a non-negative integer and let r , k be positive integers. Then

The generating function for C k , r ⁢ ( n ) , n ⩾ 0 , is

Proof

Let { i 1 , … , i r } be a set of distinct positive numbers. The set of partitions of 𝑛 having k ⁢ i 1 , … , k ⁢ i r as parts is in bijection with the set of partitions of n − k ⁢ ( i 1 + ⋯ + i r ) (where as usual the set of partitions of a negative integer is taken to be the empty set). Similarly, the set of partitions of 𝑛 with each of the parts i 1 , … , i r occurring at least 𝑘 times is in bijection with the set of partitions of n − k ⁢ ( i 1 + ⋯ + i r ) . Thus the generating functions of both C k , r ⁢ ( n ) and of D k , r ⁢ ( n ) equal Q r ⁢ ( t k ) ⁢ P ⁢ ( t ) . Now the result follows from Lemma 2.9. ∎

Let 𝑛 be a non-negative integer. If λ ⊢ n , let e ⁢ ( λ ) denote the number of ordered pairs ( m , m ′ ) of positive integers such that 𝑚 and m ′ are parts of 𝜆, m ≠ m ′ , λ m ⩾ 2 , and m ′ is even. Let E ⁢ ( n ) = ∑ λ ⊢ n e ⁢ ( λ ) .

The generating function for E ⁢ ( n ) , n ⩾ 0 is

Proof

Let 𝑚 be a positive integer and let m ′ = 2 ⁢ m ′′ ≠ m be an even positive integer. The set of partitions of 𝑛 having both 𝑚 and m ′ as parts and with the multiplicity of 𝑚 as a part being at least 2 is in bijection with the partitions of n − 2 ⁢ ( m + m ′′ ) . Thus the generating function for E ⁢ ( n ) , n ⩾ 0 , is R ⁢ ( t 2 ) ⁢ P ⁢ ( t ) , where R ⁢ ( t ) = ∑ i ⩾ 0 ∞ R i ⁢ t i and R i equals the number of ordered pairs ( m , m ′′ ) of positive integers such that i = m + m ′′ and m ≠ 2 ⁢ m ′′ . We have

Thus

proving the result. ∎

Let 𝑛 be a positive integer. We have an isomorphism of graded vector spaces

Proof

The conjugacy classes of S n correspond to partitions of 𝑛. If

then an element 𝑔 of cycle type 𝜆 has as centraliser the direct product of wreath products

Applying the Künneth formula to this direct product of groups yields the result. ∎

Let ℓ, 𝑚 be positive integers. We have an isomorphism of graded vector spaces

Proof

This follows from Nakaoka [9, Theorem 3.3].∎

For ℓ a positive integer, we have H 0 ⁢ ( S ℓ , M ℓ ) ≅ F p .

Proof

By Frobenius reciprocity, H 0 ⁢ ( S ℓ , M ℓ ) ≅ H 0 ⁢ ( S ℓ − 1 , F p ) ≅ F p . ∎

Let ℓ, 𝑚 be positive integers. As modules for S ℓ , we have

Proof

If p ∤ m , then the cohomology of ( Z / m ) m with coefficients in F p vanishes in all positive degrees. If p ∣ m , then Hom ⁢ ( Z / m , F p ) ≅ Hom ⁢ ( Z / p , F p ) ≅ F p . The result follows by looking at the action of S ℓ on the two sides of the isomorphism

Let ℓ, 𝑚 be positive integers. We have

Proof

This follows from Lemma 4.1 and Lemma 4.2. ∎

Let 𝑛 be a positive integer and 𝜆 a partition of 𝑛. We have

Proof

Only summands with λ m ⩾ 1 contribute to the left side. It follows from Lemma 4.3 that the left side counts the number of parts of 𝜆 of distinct lengths divisible by 𝑝, and this is g p , 1 ⁢ ( λ ) .∎

Let 𝑛 be a positive integer and 𝜆 a partition of 𝑛. We have

Proof

Only summands with λ m ⩾ 1 contribute to the left side. For ℓ a positive integer, we have H 1 ⁢ ( S ℓ , F p ) = Hom ⁢ ( S ℓ , F p ) = 0 for p ⩾ 3 or ℓ = 1 , and dim F 2 H 1 ⁢ ( S ℓ , F 2 ) = dim F 2 Hom ⁢ ( S ℓ , F 2 ) = 1 if ℓ ⩾ 2 . Thus the left side in the statement is zero for p ⩾ 3 . For p = 2 , this counts the number of distinct part lengths of 𝜆 that appear at least twice, and this is g 1 , 2 ⁢ ( λ ) . ∎

Let 𝑛 be a positive integer. The dimension of the degree one Hochschild cohomology of the group algebra of the symmetric group S n on 𝑛 letters is given by

Proof

Let 𝜆 be a partition of 𝑛. Note that the degree one part of the tensor product on the right side in Proposition 3.1 corresponding to the summand indexed by 𝜆 is isomorphic to

Fix an integer 𝑚 such that 1 ⩽ m ⩽ n . Applying Proposition 3.2 in degree one, we have

Taking the sum over all partitions of 𝑛 and all 𝑚 such that 1 ⩽ m ⩽ n , we obtain

Adding up dimensions, using Lemma 4.4 and Lemma 4.5 yields the result. ∎

For any positive integer 𝑛, we have

Proof

This follows from combining Theorem 4.6 and Theorem 2.6.∎

Proof of Theorem 1.2 (1.3)

Combining Corollary 4.7 with Proposition 2.3 yields the result. ∎

Let ℓ be a positive integer. We have

Proof

The module Λ 2 ⁢ ( M ℓ ) is zero for ℓ = 1 . For ℓ ⩾ 2 , this module is induced from the trivial tensor sign representation F p ⊗ sgn of S ℓ − 2 × S 2 , so by Frobenius reciprocity, we have

If p ≠ 2 , the module F p ⊗ sgn is not in the principal block, so there is no cohomology in any degree. If p = 2 , F p ⊗ sgn is the trivial module, and the computation is straightforward. ∎

Let ℓ, 𝑚 be positive integers. As an F p ⁢ S ℓ -module, we have

Proof

For p ∤ m , this is clear, so suppose that p ∣ m . If 𝑝 is odd, the cohomology is a tensor product of an exterior algebra on 𝑚 generators in degree one, tensored with a polynomial ring on their Bocksteins in degree two. So, in degree two, we have the polynomial generators, which give a copy of M ℓ , and the products of two exterior generators, which give a copy of Λ 2 ⁢ ( M ℓ ) . If p = 2 and 4 ∣ m , the computation is the same. If p = 2 and m ≡ 2 ( mod 4 ) , then the cohomology is a polynomial ring on the degree one generators, so in degree two, we have S 2 ⁢ ( M ℓ ) . As a module for S ℓ , this is isomorphic to M ℓ ⊕ Λ 2 ⁢ ( M ℓ ) . The squares of degree one generators give a copy of M ℓ , and the products of distinct degree one generators give a copy of Λ 2 ⁢ ( M ℓ ) . ∎

Let ℓ, 𝑚 be positive integers. We have

Proof

By Lemma 5.2, we get zero if p ∤ m and two terms when p ∣ m . The computation of H 0 ⁢ ( S ℓ , M ℓ ) is given in Lemma 4.1, and that of H 0 ⁢ ( S ℓ , Λ 2 ⁢ ( M ℓ ) ) is given in Lemma 5.1. ∎

Let ℓ be a positive integer. We have

Proof

By Frobenius reciprocity, we have H 1 ⁢ ( S ℓ , M ℓ ) ≅ H 1 ⁢ ( S ℓ − 1 , F p ) . ∎

Let ℓ, 𝑚 be positive integers. We have

Proof

By Lemma 4.2, H 1 ⁢ ( ( Z / m ) ℓ , F p ) is equal to M ℓ if p ∣ m and zero otherwise. Now use Lemma 5.4. ∎

Let ℓ be a positive integer. We have

and H 2 ⁢ ( S ℓ , F p ) = 0 for p ⩾ 3 .

Proof

The group H 1 ⁢ ( S ℓ , C × ) is trivial for ℓ = 1 and isomorphic to Z / 2 for ℓ ⩾ 2 . The Schur multiplier H 2 ⁢ ( S n , C × ) is trivial for ℓ ⩽ 3 and isomorphic to Z / 2 for ℓ ⩾ 4 . Raising complex numbers to their 𝑝-th powers induces a surjective group homomorphism φ : C × → C × with cyclic kernel of order 𝑝, which we identify with the additive group of F p . The associated long exact cohomology sequence in low degrees takes the form

The maps induced by 𝜑 on H 1 ⁢ ( S ℓ , C × ) and on H 2 ⁢ ( S ℓ , C × ) are isomorphisms if 𝑝 is odd and trivial if p = 2 . The result follows in all cases from the exactness of the above sequence. ∎

The dimension of the degree two Hochschild cohomology of the symmetric groups is given by

Proof

By Proposition 3.1,

where the indices { m , m ′ } run over unordered pairs of distinct positive integers between 1 and 𝑛.

Using Proposition 3.2, we have

Using Lemmas 5.3, 5.5 and 5.6, we get that, for p ⩾ 3 , the first term on the right contributes G p , 1 ⁢ ( n ) , the second and third contribute zero to the sum first sum on the right of equation (5.8). For p = 2 , the first term contributes G 2 , 1 ⁢ ( n ) + G 2 , 2 ⁢ ( n ) , the second contributes G 2 , 3 ⁢ ( n ) , and the third contributes G 1 , 2 ⁢ ( n ) + G 1 , 4 ⁢ ( n ) .

Next we consider the contribution of the second sum on the right-hand side of equation (5.8). Suppose first that p ⩾ 3 . As in the proof of Theorem 4.6, for any positive integer 𝑚,

Thus the contribution of a partition λ ⊢ n to the second sum of equation (5.8) equals c p , 2 ⁢ ( λ ) , and the second sum equals C p , 2 ⁢ ( n ) . This completes the proof for p ⩾ 3 . Now suppose that p = 2 . Again as in the proof of Theorem 4.6, for any positive integer 𝑚,

Thus the contribution of a partition λ ⊢ n to the second sum of equation (5.8) equals

and the second sum equals C 2 , 2 ⁢ ( n ) + E ⁢ ( n ) + D 2 , 2 ⁢ ( n ) , as required. ∎

For any positive integer 𝑛, we have

Proof

This follows from combining Theorem 5.7 with Theorem 2.6 and the first equality in Theorem 2.11. ∎

Proof of Theorem 1.2 (1.4)

Applying Proposition 2.3, Theorem 2.11 and Theorem 2.13 to the terms in the formula in Corollary 5.9, we obtain

Simplifying the expression for p = 2 completes the proof.∎