Nichols algebras associated to the transpositions of the symmetric group are twist-equivalent

Using the theory of covering groups of Schur we prove that the two Nichols algebras associated to the conjugacy class of transpositions in S_n are equivalent by twist and hence they have the same Hilbert series. These algebras appear in the classification of pointed Hopf algebras and in the study of quantum cohomology ring of flag manifolds.


Introduction
Nichols algebras play a fundamental role in the classification of finitedimensional pointed Hopf algebras over C. They are graded Hopf algebras in the category of Yetter-Drinfeld modules over a Hopf algebra H, and they are uniquely determined by V , the homogeneous component of B(V ) of degree one.
Let H be the group algebra of a finite group G. In the study of Nichols algebras a basic question is to describe those Yetter-Drinfeld modules V over H for which B(V ) is finite-dimensional. Whereas deep results were found for the case where G is abelian, [5,16,17], the situation is widely unknown for non-abelian groups G.
The first examples of finite-dimensional pointed Hopf algebras with nonabelian coradical appeared in [21], as bosonizations of Nichols algebras related to the transpositions in S 3 and S 4 . The analogous Nichols algebra over S 5 was computed by Graña, see [14]. These Nichols algebras are computed from the conjugacy class of transpositions and a 2-cocycle (cocycle for short) associated to this conjugacy class. The cocycles arise from a cohomology theory defined for racks (see for example [3,9,13]). In [2,Theorem 1.1] it is proved that for all n ∈ N, n ≥ 4, there are precisely two rack 2-cocycles associated to the conjugacy class of transpositions in S n that might have finite-dimensional Nichols algebras. Explicitly, one of these cocycles is the constant cocycle −1. The other one is the cocycle given by for all transpositions σ and τ = (i j) with i < j. For all n ∈ {4, 5} the Nichols algebra associated to the conjugacy class of transpositions in S n and 2010 Mathematics Subject Classification. 16T05; 16T30; 17B37. This work was partially supported by CONICET.
any of the two cocycles −1, χ is finite-dimensional. Moreover, both of these algebras have the same Hilbert series. It is not known whether these algebras are finite-dimensional for n > 5. The main result of this work is to connect these two algebras by twisting the cocycle. More precisely, we prove that the constant cocycle −1 and χ are equivalent by twist. This gives an affirmative answer to a question due to Andruskiewitsch, see [1,Question 7]. However, the problem arose already earlier in the literature. For example, in the last paragraph of [19], Majid discusses the relationship between these two algebras and the related quadratic algebras. To reach our main result, we use the existence of projective representations of S n . Projective representations of S n were originally studied by Schur in 1911, see [22] for an English translation of his fundamental paper about this subject. As a corollary of our result we obtain that for all n ≥ 4 both Nichols algebras associated to the conjugacy class of transpositions of S n have the same Hilbert series.
We recall briefly another application for Nichols algebras which may have connections with the main result of this work. In [8], Borel identified the cohomology ring of a flag manifold with S W , the algebra of coinvariants of the associated Coxeter group W . This admits certain divided-difference operators which create classes of Schubert manifolds. In [11], Fomin and Kirillov introduced a new model for the Schubert calculus of a flag manifold, realizing S W as a commutative subalgebra of a noncommutative quadratic algebra E W , when W is a symmetric group. In [6], Bazlov proved that Nichols algebras provide the correct setting for this model for Schubert calculus on a flag manifold. It is an open problem whether the Nichols algebra associated to χ coincides with the quadratic algebra E W [19,21].

Preliminaries
2.1. Racks and cohomology. We briefly recall basic facts about racks, see [3] for more information and references.
A rack is a pair (X, ⊲) where X is a non-empty set and ⊲ : X × X → X is a function, such that the map x → i ⊲ x is bijective for all i ∈ X, and In particular, the conjugacy class of transpositions in S n is a rack; it will be denoted by X n .
In this work we are interested only in racks which can be realized as a finite conjugacy class of a group. Let X be such a rack. A map q : X × X → C × is a 2-cocycle if and only if q x,y⊲z q y,z = q x⊲y,x⊲z q x,z for all x, y, z ∈ X. We write Z 2 R (X, C × ) for the set of all rack 2-cocycles.
for all x, y ∈ X. Since ∼ is an equivalence relation and Z 2 R (X, C × ) is stable under ∼ it is possible to define the second rack cohomology group as All these notions are based on the abelian cohomology theory of racks proposed independently in [9], [13]. For more details about cohomology theories of racks see [3, §4].
2.2. Nichols algebras. We refer to [4] for an introduction to Yetter-Drinfeld modules and Nichols algebras.
Let n ∈ N. We recall the well-known presentation of the braid group B n by generators and relations. The group B n has generators σ 1 , . . . , σ n−1 and relations There exists a canonical projection B n → S n that admits the so-called Matsumoto section µ : S n → B n such that µ ((i i + 1)) = σ i . This section satisfies the following: µ(xy) = µ(x)µ(y) for any x, y ∈ S n such that If c is a solution of the braid equation, we say that (V, c) is a braided vector space. A solution of the braid equation induces a representation By [3, Theorem 4.14], Yetter-Drinfeld modules over group algebras can also be studied in terms of racks and rack 2-cocycles. Therefore we are interested in Nichols algebras of braided vector spaces arising from racks and 2-cocycles.
Let (X, ⊲) be a rack and let q ∈ Z 2 R (X, C × ). We consider V = CX, the vector space with basis x ∈ X, and define c : Then c is a solution of the braid equation. The Nichols algebra associated to the pair (X, q) is the Nichols algebra of the braided vector space (V, c). This algebra will be denoted by B(X, q).
Recall that X n is defined as the rack associated to the conjugacy class of transpositions in S n . In [2, Theorem 1.1] it is proved that there are two rack 2-cocycles associated to X n that might have a finite-dimensional Nichols algebra. One is the constant 2-cocycle −1. The other is the 2-cocycle χ given by Equation (1).
Remark 2.2. It can be checked directly that the 2-cocycles −1 and χ associated to the rack X 3 are cohomologous. Then the Nichols algebras B(X 3 , χ) and B(X 3 , −1) are isomorphic and hence they have the same Hilbert series.
Example 2.3. The Nichols algebras B(X 4 , −1) and B(X 4 , χ) have both dimension 576. In both cases the Hilbert series is These algebras appeared first in [11,21]. For more information about these algebras see [ Example 2.4. The Nichols algebras B(X 5 , −1) and B(X 5 , χ) have both dimension 8294400. In both cases the Hilbert series is These algebras were first computed by Graña, [14]. For more information about these algebras see [

2.3.
Twisting. In [1, Section 3.4] it is shown how to relate two rack 2cocycles by a twisting in such a way that some properties of the corresponding Nichols algebras are preserved. This method is based on the twisting method of [10] and its relationship with the bosonization given in [20].
Let X be a subrack of a conjugacy class of a group G. Let q be a rack 2-cocycle on X and let φ be a group 2-cocycle on G. Define q φ : X ×X → C × by (2) q φ x,y = φ(x, y)φ(x ⊲ y, x) −1 q x,y for x, y ∈ X.
Remark 2.5. Let X be a rack and q ∈ H 2 R (X, C × ). For a map φ : X × X → C × define q φ by Equation (2). Then q φ is a rack 2-cocycle if and only if for any x, y, z ∈ X. Thus, if X is a subrack of a group G and φ is a group 2cocycle, φ ∈ Z 2 (G, C × ), then φ| X×X satisfies Equation  Definition 2.7. The 2-cocycles q and q ′ on X are equivalent by twist if there exists φ : X × X → C × such that q ′ = q φ as in (2). 3. The Schur cover of S n 3.1. Projective representations and covering groups. We review some aspects of Schur's theory of projective representations and construct the Schur cover of S n . See [7,18,22] for details.
A projective representation of a finite group G is a group homomorphism G → PGL(V ). Equivalently, such a representation may be viewed as a map f : for all x, y ∈ G and suitable scalars φ(x, y) ∈ C × . The map G × G → C × , (x, y) → φ(x, y), is called a factor set. The associativity of the group GL(V ) implies the 2-cocycle condition of the factor set φ: for all x, y, z ∈ G. Two projective representations ρ 1 : G → GL(V 1 ) and ρ 2 : for all x ∈ G. Two factor sets φ and φ ′ are equivalent if they differ only by a factor b x b y /b xy for some b : G → C × . The Schur multiplier of G is the abelian group of factor sets modulo equivalence. It is isomorphic to the second cohomology group H 2 (G, C × ).
Recall that a central extension of G is a pair (E, p), where p : E → G is a surjective group homomorphism such that ker p is contained in the center of the group E. Schur proved that every finite group G has a central extension (E, p) with the property that every projective representation ρ of G lifts to an ordinary representationρ of E such that the diagram

commutes.
There exist extensions with ker p ≃ H 2 (G, C × ). Moreover, H 2 (G, C × ) is the unique minimal possibility for ker p. These minimal central extensions of G are called Schur covering groups of G.
Theorem 3.1. Given n ≥ 4, define the group T n as follows Then T n is a Schur covering group of S n . Therefore, there exists a central extension Remark 3.2. Let t ∈ T n . For any σ ∈ S n we have that p −1 (σ) = {σ,σz}.
Since the involution z is a central element of T n , the group S n acts on T n by conjugation: σ ⊲ t =σt(σ) −1 = (σz)t(σz) −1 . Therefore it is possible to write the conjugation in T n as σ ⊲ t = σtσ −1 , where t ∈ T n and σ ∈ S n . Definition 3.3. For i, j ∈ N such that 1 ≤ i, j ≤ n, i = j, let [i j] be an element of T n defined inductively as Proof. Multiplying both sides by z if needed, we may assume that i < j. If {k, k + 1} ∩ {i, j} = ∅ then the claim follows from [22,Paragraph 6,III]. If k = i − 1 then the claim follows from Definition 3.3. The case k = i follows from the case k = i − 1 by applying s i−1 . Since s j ⊲ t j−1 = s j−1 ⊲ t j , a straightforward computation settles the case k = j. Finally, the case k = j − 1 follows from the case k = j by applying s j . Proposition 3.5. Let l ∈ N, σ = s i 1 s i 2 · · · s i l ∈ S n and i, j ∈ {1, ..., n}.
Proof. Follows from Lemma 3.4 by induction on l.
3.3. Nichols algebras over symmetric groups. Recall that X n is the rack of transpositions in S n . There exist two rack 2-cocycles that we want to consider. One of these rack 2-cocycles is the constant cocycle −1. The other one is the 2-cocycle given by Equation (1).
Lemma 3.7. There exists a section s : S n → T n such that if τ = (i j), i < j, then for all σ.
Proof. By Theorem 3.1 there exists a central extension where A = z . Take any set-theoretical sections : S n → T n such that s(id) = 1 and define a map s : S n → T n by (6) s(π) = s(π) if π / ∈ X n , [i j] if π = (i j) ∈ X n , with i < j.
Then ps = id and s(id) = 1. Since σ ∈ X n , the length of σ is 1. Remark 3.2 and Proposition 3.5 imply that Hence the claim follows. where A = z . Let s : S n → T n be the section of Lemma 3.7 and let φ(x, y) ∈ A be defined by the equation s(x)s(y) = i(φ(x, y))s(xy).