ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.27877 J. Algebra Comb. Discrete Appl. 3(3) • 195–200 Received: 04 November 2015 Accepted: 08 April 2016 Journal of Algebra Combinatorics Discrete Structures and Applications Quasisymmetric functions and Heisenberg doubles Research Article Jie Sun Abstract: The ring of quasisymmetric functions is free over the ring of symmetric functions. This result was previously proved by M. Hazewinkel combinatorially through constructing a polynomial basis for quasisymmetric functions. The recent work by A. Savage and O. Yacobi on representation theory provides a new proof to this result. In this paper, we proved that under certain conditions, the positive part of a Heisenberg double is free over the positive part of the corresponding projective Heisenberg double. Examples satisfying the above conditions are discussed. 2010 MSC: 16T05, 05E05 Keywords: Quasisymmetric function, Heisenberg double, Tower of algebras, Hopf algebra, Fock space 1. Introduction Symmetric functions are formal power series which are invariant under every permutation of the in- determinates ([14]). Let Sym denotes the ring of symmetric functions over integers. Then the elementary symmetric functions form a polynomial basis of Sym. The existence of comultiplication and counit gives Sym a Hopf algebra structure. It’s well-known (see [4]) that as Hopf algebras Sym is isomorphic to the Grothendieck group of the abelian category C[Sn]-mod, where Sn is the n-th symmetric group. Among different generalizations of symmetric functions, there are noncommutative symmetric func- tions and quasisymmetric functions. As an algebra, NSym is the free algebra Z〈h1,h2, · · · 〉. As a Hopf algebra, NSym is isomorphic to the Grothendieck group of the abelian category Hn(0)-pmod ([3], [9], [15]), where Hn(0) is the 0-Hecke algebra of degree n. The ring of quasisymmetric func- tions QSym ⊂ Z[[x1,x2, · · · ]] consists of shift invariant formal power series of bounded degrees. Let Comp(n) denote the set of compositions of n. Then the monomial quasisymmetric functions Mα, where α ∈ ⋃ n∈N Comp(n), form an additive basis for QSym ([14]). As Hopf algebras, NSym and QSym are dual to each other under the bilinear form 〈hα,Mβ〉 = δα,β. Polynomial freeness of QSym was conjectured by E. Ditters in 1972 in his development of the theory of formal groups ([2]). Ditters conjecture was proved by M. Hazewinkel ([7], [6]) combinatorially through Jie Sun; Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA (email: sjie@mtu.edu). 195 J. Sun / J. Algebra Comb. Discrete Appl. 3(3) (2016) 195–200 constructing a polynomial basis for quasisymmetric functions. The explicit basis constructed by M. Hazewinkel contains the elementary symmetric functions, hence QSym is free over Sym. From the representation theory point of view, A. Savage and O. Yacobi [12] provide a new proof to the freeness of QSym over Sym. For each dual pair of Hopf algebras (H+,H−), one can construct the Heisenberg double h = h(H+,H−) of H+. The algebra h has a natural representation on H+, called the Fock space representation F. In [12] it is proved that any representation of h generated by a lowest weight vacuum vector is isomorphic to F. As an application of this Stone-von Neumann type theorem to h(QSym,NSym), A. Savage and O. Yacobi gave a different proof that QSym is free as a Sym-module (Proposition 9.2 in [12]). In this paper, after review some basic definitions and examples, we proved our main result Theorem 3.2 which says that under certain conditions, the positive part of a Heisenberg double is free over the positive part of the corresponding projective Heisenberg double. Examples satisfying the conditions of Theorem 3.2 are discussed. 2. Definitions and examples In this section, we will recall some basic definitions and examples. Most of the definitions can be found in [12]. Fix a commutative ring K. We use Sweedler notation 4(a) = ∑ (a) a(1) ⊗ a(2) for coproducts. Definition 2.1. (Dual pair). We say that (H+,H−) is a dual pair of Hopf algebras if H+ and H− are both graded connected Hopf algebras and H± is graded dual to H∓ via a perfect Hopf pairing 〈·, ·〉 : H+ ×H− → K. From a dual pair of Hopf algebras, one can construct the Heisenberg double. Definition 2.2. (The Heisenberg double, [13]). We define h(H+,H−) to be the Heisenberg double of H+. As K-modules h(H+,H−) ∼= H+ ⊗ H− and we write a]x for a ⊗ x, a ∈ H+, x ∈ H−, viewed as an element of h(H+,H−). Multiplication is given by (a]x)(b]y) = ∑ (x) a Rx∗ (1) (b)]x(2)y =∑ (x),(b)〈x(1),b(2)〉ab(1)]x(2)y, where Rx∗ (1) (b) is the left-regular action of H− on H+. For the Heisenberg double h = h(H+,H−), there is a natural representation, called the Fock space representation. Definition 2.3. (Fock space representation). The algebra h has a natural representation on H+ given by (a]x)(b) = a Rx∗(b), a,b ∈ H+,x ∈ H−, which is called the lowest weight Fock space representation of h and is denoted by F = F(H+,H−). It is generated by the lowest weight vacuum vector 1 ∈ H+. Examples of the Heisenberg double include the usual Heisenberg algebra and the quasi-Heisenberg algebra. When we take H+ = H− = Sym, the Heisenberg double h = h(Sym,Sym) is the usual Heisenberg algebra. Indeed, we can take p1,p2, · · · to be the power sums in H+ and p∗1,p∗2, · · · to be the power sums in H−, then the multiplication in h = h(Sym,Sym) gives the usual presentation of the Heisenberg algebra: pmpn = pnpm, p ∗ mp ∗ n = p ∗ np ∗ m, p ∗ mpn = pnp ∗ m + mδm,n. When we take H + = QSym and H− = NSym, the Heisenberg double h = h(QSym,NSym) is the quasi-Heisenberg algebra. The natural action on QSym is the Fock space representation. Both the Heisenberg algebra and the quasi-Heisenberg algebra can be regarded as the Heisenberg double associated to a tower of algebras. We now recall the definition of tower of algebras in the following. Definition 2.4. Let A = ⊕n∈NAn be a graded algebra over a field F with multiplication ρ : A⊗A → A. Then A is called a tower of algebras if the following conditions are satisfied: (TA1) Each graded piece An, n ∈ N, is a finite dimensional algebra (with a different multiplication) with a unit 1n. We have A0 = F. 196 J. Sun / J. Algebra Comb. Discrete Appl. 3(3) (2016) 195–200 (TA2) The external multiplication ρm,n : Am ⊗An → Am+n is a homomorphism of algebras for all m,n ∈ N (sending 1m ⊗1n to 1m+n). (TA3) We have that Am+n is a two-sided projective Am ⊗ An-module with the action defined by a · (b⊗ c) = aρm,n(b⊗ c) and (b⊗ c) ·a = ρm,n(b⊗ c)a, for all m,n ∈ N, a ∈ Am+n, b ∈ Am, c ∈ An. (TA4) For each n ∈ N, the pairing 〈·, ·〉 : K0(An) × G0(An) → Z, given by 〈[P ], [M]〉 = dimF HomAn(P,M), is perfect. (Note this condition is automatically satisfied if F is an algebraically closed field.) In the above definition the notation G0(An) = K0(An-mod) denotes the Grothendieck group of the abelian category An-mod, and K0(An) = K0(An-pmod) denotes the Grothendieck group of the abelian category An-pmod. For the rest of this section we assume that A is a tower of algebras and let G(A) = ⊕n∈NG0(An) and K(A) = ⊕n∈NK0(An). We have a perfect pairing 〈·, ·〉 : K(A) ×G(A) → Z given by 〈[P ], [M]〉 = dimF HomAn(P,M) if P ∈ An- pmod and M ∈ An-mod for some n ∈ N, and 0 otherwise. Definition 2.5. (Strong tower of algebras) A tower of algebras A is strong if induction is conjugate right adjoint to restriction and a Mackey-like isomorphism relating induction and restriction holds. For the technical definition of the Mackey-like isomorphism in Definition 2.5 we refer the reader to [12] (Definition 3.4 and Remark 3.5). Definition 2.6. (Dualizing tower of algebras) A tower of algebras A is dualizing if K(A) and G(A) are dual pair Hopf algebras. Strong dualizing towers of algebras categorify the Heisenberg double ([1]) and its Fock space repre- sentation ([12] Theorem 3.18). To a dualizing tower of algebras, we can define the associated Heisenberg double. Definition 2.7. (Heisenberg double associated to a tower) Suppose A is a dualizing tower of algebras. Then h(A) = h(G(A),K(A)) is the associated Heisenberg double and F(A) = F(G(A),K(A)) is the Fock space representation of h(A). When An is the Hecke algebra at a generic value of q, the associated Heisenberg double is the usual Heisenberg algebra. When An is the 0-Hecke algebra, the associated Heisenberg double is the quasi-Heisenberg algebra. Fix a dualizing tower of algebras A. For each n ∈ N, An-pmod is a full subcategory of An-mod. The inclusion functor induces the Cartan map K(A) → G(A). Let Gproj(A) denote the image of the Cartan map. Let H− = K(A), H+ = G(A), H+proj = Gproj(A), h = h(A), F = F(A). Proposition 3.12 in [12] showed that H+proj is a subalgebra of H + that is invariant under the left-regular action of H−. We next recall the projective Heisenberg double and its Fock space in the following. Definition 2.8. (The projective Heisenberg double hproj) The subalgebra hproj = hproj(A) := H + proj]H − (the subalgebra of h generated by H+proj and H −) is called the projective Heisenberg double associated to A. Definition 2.9. (Fock space Fproj of hproj) The action of the algebra hproj on H+proj is called the lowest weight Fock space representation of hproj and is denoted by Fproj = Fproj(A). It is generated by the lowest weight vacuum vector 1 ∈ H+proj. In [12] a Stone-von Neumann type theorem (Theorem 2.11 in [12]) was proved. A consequence of this theorem tells that any representation of hproj generated by a lowest weight vacuum vector is isomorphic to Fproj (see Proposition 3.15 in [12]). 197 J. Sun / J. Algebra Comb. Discrete Appl. 3(3) (2016) 195–200 3. Main result In this section, we will first prove a lemma about the complete reducibility of an hproj-module generated by a finite set of lowest weight vacuum vectors. Then we will prove our main result which says that under certain conditions, the positive part of a Heisenberg double is free over the positive part of the corresponding projective Heisenberg double. We will use the notations from section 2. Lemma 3.1. Suppose V is an hproj-module which is generated as an hproj-module by a finite set of lowest weight vacuum vectors. Then V is a direct sum of copies of lowest weight Fock space Fproj. Proof. Let {vi}i∈I denote a finite generating set of V consisting of lowest weight vacuum vectors. Suppose that I has minimal cardinality. Suppose that Zvi∩Zvj 6= {0}, for some i 6= j. Then nivi = njvj for some ni,nj ∈ Z. Let m = gcd(ni,nj). Then there exists ai,aj ∈ Z such that m = aini + ajnj. Let w = ajvi + aivj. Then w is a lowest weight vacuum vector. Calculation shows that ni m w = 1 m (ajnivi + ainivj) = 1 m (ajnjvj + ainivj) = vj, nj m w = 1 m (ajnjvi + ainjvj) = 1 m (ajnjvi + ainivi) = vi. Therefore, {vk}k∈I\{i,j} ∪{w} is also a generating set of V consisting of lowest weight vacuum vectors. This contradicts the minimality of the cardinality of I. Thus Zvi ∩Zvj = {0}, for all i 6= j. By Proposition 3.15(c) in [12], any representation of hproj generated by a lowest weight vacuum vector is isomorphic to Fproj. Thus hproj ·vi ∼= Fproj as hproj-modules. By Propostion 3.15(a) in [12], the only submodules of Fproj are those submodules of the form nFproj for n ∈ Z. Therefore hproj·vi∩hproj·vj = {0} for i 6= j. The complete reducibility of V follows. Theorem 3.2. Let A be a dualizing tower of algebras. Suppose there exists an increasing filtration {0}⊂ H+proj = (H +)(0) ⊂ (H+)(1) ⊂ (H+)(2) ⊂ ··· of hproj-submodules of H+ such that (H+)(n)/(H+)(n−1) is generated by a finite set of vacuum vectors, then H+ is free as an H+proj-module. Proof. Let Vn = (H+)(n)/(H+)(n−1), then by Lemma 3.1, Vn = ⊕v∈LnH + proj · v, where Ln is some collection of vacuum vectors in Vn. Consider the short exact sequence 0 → (H+)(n−1) → (H+)(n) → Vn → 0. Since Vn is a free H + proj-module, the above short exact sequence split. By induction on n, we know that all (H+)(n) (n ∈ N) is free over H+proj. Thus we can choose nested sets of vectors in H +: L̃0 ⊂ L̃1 ⊂ L̃2 ⊂ ··· such that for each n ∈ N, we have (H+)(n) = ⊕ ṽ∈L̃n H+proj ·ṽ. Let L̃ = ⋃ n∈N L̃n, then H + = ⊕ v∈L̃H + proj ·v and H+ is free over H+proj. The main ideas used in proving Lemma 3.1 and Theorem 3.2 follow from Lemma 9.1 and Proposition 9.2 in [12]. Observing that the existence of a special filtration of hproj-submodules of H+ is the key to the polynomial freeness of H+ over H+proj, we see that Theorem 3.2 generalizes the result of Proposition 9.2 in [12] (see the discussion of Example 4.1 in section 4). 198 J. Sun / J. Algebra Comb. Discrete Appl. 3(3) (2016) 195–200 4. Applications In this section, we will discuss some applications of our main theorem. Example 4.1. (Tower of 0-Hecke algebras) Let A = ⊕n∈NHn(0), where Hn(0) is the 0-Hecke algebra of degree n. Then H+ = G(A) = QSym and H− = K(A) = NSym. The associated Heisenberg double is the quasi-Heisenberg algebra q = h(QSym,NSym). The projective quasi-Heisenberg algebra qproj is the subalgebra generated by H + proj = Sym ⊂ QSym and H− = NSym. For n ∈ N, let QSym(n) := ∑ l(α)≤n qproj ·Mα, where α ∈ ⋃ n∈N Comp(n) and l(α) is the number of nonzero parts of α. Then {0}⊂ Sym = QSym (0) ⊂ QSym(1) ⊂ QSym(2) ⊂ ··· defines an increasing filtration of qproj-submodules of QSym. For α ∈⋃ n∈N Comp(n) such that l(α) = n, we have Rh∗m(Mα) ∈ QSym (n−1) for any m > 0. So all Mα with l(α) = n are lowest weight vacuum vectors in the quotient Vn = QSym (n)/QSym(n−1) and generate Vn. The condition of Theorem 3.2 is satisfied. Therefore, H+ = QSym is free over H+proj = Sym. This recovers Proposition 9.2 in [12]. Example 4.2. (Tower of 0-Hecke-Clifford algebras) Let A = ⊕n∈NHCln(0), where HCln(0) is the 0-Hecke-Clifford algebra of degree n. Then H+ = G(A) and H− = K(A) form a dual pair of Hopf algebras. There are two main ideas used in [10] (section 3.3) to prove that (G(A),K(A)) is a dual pair of Hopf algebras. First, the Mackey property of 0-Hecke- Clifford algebras guarantees that G(A) and K(A) are Hopf algebras. Second, it is shown that HCln(0) is a Frobenius superalgebra which satisfies the conditions of Proposition 6.7 in [11]. The associated Heisenberg double is h(G(A),K(A)) and the corresponding projective Heisenberg double is hproj. Here H+ = Peak∗ is the space of peak quasisymmetric functions. Let θ : QSym → Peak∗ be the descent-to- peak map and let Nα = θ(Mα). Then H+ is spanned by Nα, where α ∈ ⋃ n∈N Comp(n). For n ∈ N, let (Peak∗)(n) := ∑ l(α)≤n hproj ·Nα. Then {0} ⊂ H+proj = (Peak ∗)(0) ⊂ (Peak∗)(1) ⊂ (Peak∗)(2) ⊂ ··· defines an increasing filtration of hproj-submodules of Peak ∗. We know that all Nα with l(α) = n are lowest weight vacuum vectors in the quotient Vn = (Peak ∗)(n)/(Peak∗)(n−1) and generate Vn. The condition of Theorem 3.2 is satisfied. Therefore, H+ = Peak∗ is free over H+proj, where H + proj is the subring of symmetric functions spanned by Schur’s Q-functions. This recovers Proposition 4.2.2 in [10]. Example 4.3. (Towers of algebras related to the symmetric groups and their Hecke algebras) In [8], the algebra HSn is defined to be the subalgebra of End(CSn) generated by both sets of operators from C[Sn] and Hn(0). This algebra HSn has interesting combinatorial properties. For examples, the dimensional formula calculated in [8] finds applications in the study of symmetric functions associated with stirling permutations [5]. The author conjectures that Theorem 3.2 applies to the tower of algebras ⊕n∈NHSn. The proof of this conjecture is the work in progress of the author in a forthcoming paper. Acknowledgment: The author would like to thank Alistair Savage for explaining the paper [12]. The author would also like to thank Rafael S. González D’León for pointing out the algebra HSn in [8]. 199 J. Sun / J. Algebra Comb. Discrete Appl. 3(3) (2016) 195–200 References [1] N. Bergeron, H. Li, Algebraic structures on Grothendieck groups of a tower of algebras, J. Algebra 321(8) (2009) 2068–2084. [2] E. J. Ditters, Curves and formal (Co)groups, Invent. Math. 17(1) (1972) 1–20. [3] G. Duchamp, D. Krob, B. Leclerc, J-Y. 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Gen. 29(22) (1996) 7337–7348. 200 http://dx.doi.org/10.1016/j.jalgebra.2008.12.005 http://dx.doi.org/10.1016/j.jalgebra.2008.12.005 http://dx.doi.org/10.1007/BF01390019 http://www.ams.org/mathscinet-getitem?mr=1373744 http://www.ams.org/mathscinet-getitem?mr=1373744 http://www.ams.org/mathscinet-getitem?mr=1373744 http://www.ams.org/mathscinet-getitem?mr=1373744 http://www.ams.org/mathscinet-getitem?mr=0506405 http://www.ams.org/mathscinet-getitem?mr=0506405 http://www.ams.org/mathscinet-getitem?mr=0506405 http://dx.doi.org/10.1006/aima.2001.2017 http://dx.doi.org/10.1006/aima.2001.2017 http://dx.doi.org/10.1090/surv/168 http://dx.doi.org/10.1090/surv/168 http://dx.doi.org/10.1090/surv/168 http://dx.doi.org/10.1023/A:1008673127310 http://dx.doi.org/10.1023/A:1008673127310 http://dx.doi.org/10.1016/j.jalgebra.2016.01.013 http://dx.doi.org/10.1016/j.jpaa.2015.03.016 http://dx.doi.org/10.1016/j.jpaa.2015.03.016 http://dx.doi.org/10.1016/j.jcta.2014.09.002 http://dx.doi.org/10.1016/j.jcta.2014.09.002 http://dx.doi.org/10.1090/conm/175 http://dx.doi.org/10.1090/conm/175 http://dx.doi.org/10.1090/conm/175 http://dx.doi.org/10.1088/0305-4470/29/22/027 http://dx.doi.org/10.1088/0305-4470/29/22/027 Introduction Definitions and examples Main result Applications References