Invariants in Non-ommutative Variables of the Symmetric and Hyperoctahedral Groups Anouk Bergeron-Brlek To cite this version: Anouk Bergeron-Brlek. Invariants in Non-ommutative Variables of the Symmetric and Hyperoctahedral Groups. Krattenthaler, hristian and Sagan, Bruce. 20th Annual International onference on Formal Power Series and Algebraic ombinatorics (FPSA 2008), 2008, Viña del Mar, hile. Discrete Mathematics and Theoretical omputer Science, DMTS Proceedings vol. AJ, 20th Annual International onference on Formal Power Series and Algebraic ombinatorics (FPSA 2008), pp.653-664, 2008, DMTS Proceedings. <hal-01185143> HAL Id: hal-01185143 https://hal.inria.fr/hal-01185143 Submitted on 19 Aug 2015 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
FPSA 2008, Valparaiso-Viña del Mar, hile DMTS proc. AJ, 2008, 653 664 Invariants in Non-ommutative Variables of the Symmetric and Hyperoctahedral Groups Anouk Bergeron-Brlek York University, Mathematics and Statistics, 4700 Keele Street, M3J 1P3, Toronto, anada Abstract. We consider the graded Hopf algebra NSym of symmetric functions with non-commutative variables, which is analogous to the algebra Sym of the ordinary symmetric functions in commutative variables. We give formulaes for the product and coproduct on some of the analogues of the Sym bases and expressions for a shuffle product on NSym. We also consider the invariants of the hyperoctahedral group in the non-commutative case and a state a few results. Résumé. Nous considérons l algèbre de Hopf graduée NSym des fonctions symétriques en variables non-commutatives, qui est analogue à l algèbre Sym des fonctions symétriques en variables commutatives. Nous donnons des formules pour le produit et coproduit sur certaines des bases analogues à celles de Sym, ainsi qu une expression pour le produit shuffle sur NSym. Nous considérons aussi les invariants du groupe hyperoctaédral dans le cas non-commutatif et énonçons quelques résultats. Keywords: invariants, symmetric function, non-commutative variables, Hopf algebra 1 Introduction Let X n = x 1, x 2,..., x n be a list of variables and denote by Q X n the algebra of polynomials in noncommutative variables with rational coefficients. We consider the algebra Q X n Sn of polynomials in Q X n which are invariant under the action of the symmetric group S n. In Bergeron et al. (2), this algebra is extended to the graded Hopf algebra N Sym of symmetric functions in non-commutative variables (not to be confused with the algebra of non-commutative symmetric functions presented in Gelfand et al. (5)). The Hopf algebra NSym is analogous to the algebra of the ordinary symmetric functions in commutative variables and appears in many recent works, see for instance (6; 7; 1; 3; 2). The analogues of the monomial, elementary, homogeneous and power sum bases are defined in Rosas and Sagan (10) and are indexed by set partitions. In Section 2, we recalled some basic facts about combinatorics of set partitions. In Section 3, we consider the Hopf algebra structure of that algebra. Section 4 contains some formulaes for the product and coproduct, and expressions for a shuffle product on N Sym. One surprising thing that we found with this algebra is that all the bases defined by Rosas and Sagan (10) that are analogues multiplicative bases in the commutative algebra are also multiplicative and freely generated in the non-commutative version. We also found that the non-commutative homogeneous basis with the shuffle product is dual to 1365 8050 c 2008 Discrete Mathematics and Theoretical omputer Science (DMTS), Nancy, France
654 Anouk Bergeron-Brlek the non-commutative monomial basis with the usual coproduct, and the non-commutative monomial basis with the shuffle product is dual to the non-commutative homogeneous basis with the usual coproduct. This is very unusual and says that the bases defined by Rosas and Sagan (10) also behave naturally under the shuffle product. Although this work is preliminary, at the end we introduce and state a few results on the invariants of the hyperoctahedral group B n in non-commutative variables X n. This algebra is indexed by the set partitions with at most n parts whose size are even. This makes clear the connection with the work of Orellana (9) on the centralizer algebra End Bn (V k ) and the invariants of B n. 2 Definitions and notations Let [n] = {1, 2,... n}. A set partition of n, denoted by A [n], is a family of disjoint nonempty subsets A 1, A 2,..., A k [n] such that A 1 A 2... A k = [n]. The subsets A i are called the parts of A and the length l(a) of A is the number of its parts. The size of a set partition A is denoted by A. To simplify the notation, we will write each part of A = {A 1, A 2,..., A k } as a word. Moreover, the parts of every set partition will be arranged by increasing value of the smallest element in the subset. Example. The set partitions of 3 are {1, 2, 3} {12, 3} {13, 2} {1, 23} {123}. Assume that the parts of a set partition A are listed in decreasing order of size, then to A can be associated a partition λ(a) = ( A 1, A 2,..., A k ) and A! = λ(a)! = λ 1!λ 2! λ k!. The partition λ(a, B) will denote the partition whose ith part is the number of parts A j such that A j B i, for 1 i l(b). Example. onsider A = {136, 25, 4, 7} and B = {12356, 47}. Then λ(a) = (3, 2, 1, 1), λ(a)! = 3!2!, λ(a, B) = (2, 2). The sign of a permutation σ = σ(1)σ(2) σ(n) is defined by sgn(σ) = ( 1) inv(σ), where inv(σ) is the number of inversions of σ, i.e. the number of pairs (σ(i), σ(j)) such that i < j and σ(i) > σ(j). The sign ( 1) A of a set partition A is the sign of any permutation obtained in the following way: the cycles of the permutation will be formed by taking the integers in each part of A. A cycle of length l has sign ( 1) l 1, so if the permutation σ = c 1... c k S n is written in cycle notation, then sgn(σ) = ( 1) l1 1... ( 1) l k 1 = ( 1) n k. Hence the sign of a set partition A [n] is ( 1) n l(a). Example. onsider A = {1325, 4}. Then ( 1) A = ( 1) 5 2 = sgn(35241) = ( 1) inv(35241) = 1. Given S [l(a)] with S = {s 1, s 2,..., s k }, we define A S = {A s1, A s2,..., A sk }. Then the standardisation of A S, st(a S ), is the set partition obtained by lowering the entries of A S and keeping the values in relative order. We will also denote by A S the set partition A restricted to the entries which are in S and A S will mean raising the entries in A so that they remain in the same relative order, with the union of all parts equal to S. Let S c denotes the complement of S.
Non-ommutative Invariants of S n and B n 655 Example. For S = {1, 3, 4} we get st({136, 25, 4, 7} S ) = {124, 3, 5}, st({136, 25, 4, 7} S ) = {12, 3}, {13, 2} S = {14, 3}. For an arbitrary set S = {s 1, s 2,..., s k }, S + n is defined to be {s 1 + n, s 2 + n,..., s k + n}. Given two set partitions A [n] and B, we can build the set partitions A B = {A 1, A 2,..., A l(a), B 1 + n, B 2 + n,..., B l(b) + n}. A set partition A [n] is called splittable if there exists non-empty set partitions B [k] and [n k] such that A = B, and is called non-splittable if it is non-empty and not splittable. If A is splittable, then A! = (A (1), A (2),..., A (d) ), where each A (i) is non-splittable, will denote the split of A. In that case, A! = A (1) A (2) A (d). Example. onsider A = {13, 25, 4}, B = {13, 2} and = {1}. Then A B = {13, 25, 4, 68, 7, 9}. There is an ordering (by refinement) on the set partitions defined as follow: We say that A B if and only if each part of A is contained in some part of B. Under this ordering, the set partitions of n form a lattice Π n. The greatest lower bound and the least upper bound of A and B will be respectively denoted by A B = {A i B j 1 i l(a), 1 j l(b)} and A B. Each part in A B is both a union only of parts of A and a union only of parts of B, with no proper subset having that property. Example. We have {136, 25, 4, 7} {167, 245, 3} = {16, 3, 25, 4, 7}, {136, 25, 4, 7} {167, 245, 3} = {1367, 245}. The set partition 0 n = {1, 2,..., n} is the minimal set partition and 1 n = {12... n} is the maximal set partition. The Möbius function of Π n is given by 1 if A = B, µ(a, B) = µ(a, ) otherwise. A <B The Kronecker delta of two set partitions A and B is defined by { 1 if A = B, δ A,B = 0 otherwise. 3 Hopf algebra structure on NSym We will concentrate here on graded-connected Hopf algebras. These are defined as a graded algebra (H, µ) with unit, a graded coalgebra structure (H, ) with counit, such that the coproduct is a morphism with respect to the algebra structure, i.e. for f, g H, τ(f g) = g f and (µ µ) (id τ id) ( ) = µ.
656 Anouk Bergeron-Brlek The symmetric group S n acts naturally on Q X n by σf(x 1, x 2,..., x n ) = f(x σ(1), x σ(2),, x σ(n) ). The algebra Q X n Sn of polynomials in Q X n which are invariant under this action can be extended to the graded algebra NSym = d 0 NSym d of symmetric functions in non-commutative variables, where NSym d is the linear span of all invariants of homogeneous degree d in the infinite x 1, x 2, x 3,... non-commuting variables. This is done via an analogous technique to the one used in the commutative version (see (8) for technical references). The analogues of the bases of the symmetric functions in commutative variables are defined as follows. For A [d], the monomial symmetric function in non-commutative variables is defined as m A = x i1 x i2... x id, (1) (i 1,i 2,...,i n) where the sum is over all sequences (i 1, i 2,..., i n ) with i a = i b if and only if a and b are in the same part in A. We refer the reader to (10) for the definition of the elementary symmetric function, the complete homogeneous symmetric function and the power sum function in non-commutative variables and changes of variables. In the last section, there is a table summarizing those ones. Written in terms of the monomial symmetric function in non-commuting variables, we have: e A = B A=0 m B, h A = B (B A)!m B, p A = B A m B. Example. m {13,2} = x 1 x 2 x 1 + x 2 x 1 x 2 + x 1 x 3 x 1 + x 3 x 1 x 3 + x 2 x 3 x 2 + x 3 x 2 x 3 + e {13,2} = m {1,2,3} + m {12,3} + m {1,23} h {13,2} = m {1,2,3} + m {12,3} + 2 m {13,2} + m {1,23} + 2 m {123} p {13,2} = m {13,2} + m {123}. In (2), they consider NSym = d 0 NSym d, where NSym d is the linear span of {m A } A [d], as a graded Hopf algebra with product and coproduct given respectively by µ : NSym d NSym k NSym d+k, : NSym d NSym k NSym d k. µ(m A m B ) := m (m A ) = m AS m AS c Π d+k 1 d 1 k = A B 4 Results on NSym The first proposition shows that the non-commutative elementary basis is multiplicative. Proposition 1 We have e A e B = e A B.
Non-ommutative Invariants of S n and B n 657 Proof: We have p A p B = p A B (see (1)) and for A [m], e A = A µ(0 m, )p (see Table 1). Thus, for a set partition B [n], e A e B = µ(0 m, )p µ(0 n, D)p D A D B = µ(0 m, )µ(0 n, D)p D. A D B Now since {G : G A B} = { D : D A B} = { D : A D B} is isomorphic to the cartesian product { : A} {D : D B} and the Möbius function µ is multiplicative in the sense that µ(a, )µ(b, D) = µ(a B, D), then e A e B = µ(0 m+n, D)p D A D B = µ(0 m+n, G)p G = e A B. G A B From Proposition 1 we get the next corollary. orollary 1 NSym is freely generated by the elements e A, where A is non splittable. Proof: Since the e A are multiplicative, we have that for A! = (A (1), A (2),..., A (k) ), e A = e A (1) A (2)... A = e (k) A (1)e A e (2) A (k). Since the e A are linearly independant, then {e A : A non-splittable} must be algebraically independant. The next corollary also follows from Proposition 1 and use the involution ω : NSym NSym defined in (10), that sends e A to h A, for all set partitions A and linear extension. orollary 2 We have h A h B = h A B. Moreover, NSym is freely generated by the elements h A, where A is non-splittable. Proof: This results from the sequence of equalities h A h B = ω(e A )ω(e B ) = ω(e A e B ) = ω(e A B ) = h A B. The coproduct on the non-commutative elementary basis and the non-commutative homogeneous basis is next given. See the Appendix for some examples. Proposition 2 For A [n], we have (e A ) = e st(a S ) e st(a S c ). S [n]
658 Anouk Bergeron-Brlek Proof: For A [n], e A = B A=0 n m B (see Table 1). Therefore ( ) (e A ) = m B = (m B ) = B A=0 n B A=0 n B A=0 n T [l(b)] m st(bt ) m st(bt c ). Note that the parts of every set partition are arranged by increasing value of the smallest element in the subset. Let S 1 = {(B, T ) B A = 0 n and T [l(b)]}, S 2 = {(, D, S) st(a S ) = 0 S, D st(a S c) = 0 S c and S [n]}. There is a bijection ϕ : S 1 S 2 defined by ϕ((b, T )) = (st(b T ), st(b T c), B T1 B T2 B T T ) with inverse ϕ 1 ((, D, S)) = ( S D S c, {i ( S D S c) i S}). Hence (e A ) = S [n] st(a S )=0 S = S [n] ( st(a S )=0 S m m m D D st(a S c )=0 S c ) ( D st(a S c )=0 S c m D ) = e st(a S ) e st(a S c ). S [n] The next lemma will be used in order to prove a formula for the coproduct on the h basis. Lemma 1 (ω ω) = ω. Proof: By applying and since ω(p A ) = ( 1 A )p A (see Table 6), we get ( ) (ω ω) (p A ) = (ω ω) p st(as ) p st(as c ) = = ω(p st(as )) ω(p st(as c )) ( 1) st(a S) p st(as ) ( 1) st(a S c ) p st(as c ). Now observe that for any S [l(a)], ( 1) st(as) ( 1) st(a S c ) = ( 1) A. Hence (ω ω) (p A ) = ( 1) A p st(as ) p st(as c ) = ( 1) A (p A ) = (( 1) A p A ) = ω(p A ).
Non-ommutative Invariants of S n and B n 659 orollary 3 For A [n], one has Proof: Let (e A ) = B, (h A ) = S [n] d B, A e B e. Then h st(a S ) h st(a S c ). (h A ) = ω(e A ) = (ω ω) (e A ) (by Lemma 1) = (ω ω)( B, Let u and v be monomials in Q X n and ( ) [ u + v ] u be the set of subsets of [ u + v ] with u elements. Let s = {s 1, s 2,..., s u } be an element of that set and let t = {t 1, t 2,..., t v } be its complement in d B, A e B e ) = B, d B, A ω(e B) ω(e ) = d B, A h B h. B, [ u + v ]. Then u sv is the monomial w such that u = w s1 w s2 w s u and v = w t1 w t2 w t v. The shuffle of the monomial u with v is then defined to be u v = u sv. Example. s ( [ u + v ] u ) x 1 x 1 x 1 x 2 = x 1 x 1 {1,2} x 1 x 2 + x 1 x 1 {1,3} x 1 x 2 + x 1 x 1 {1,4} x 1 x 2 + x 1 x 1 {2,3} x 1 x 2 + x 1 x 1 {2,4} x 1 x 2 + x 1 x 1 {3,4} x 1 x 2 = x 1 x 1 x 1 x 2 + x 1 x 1 x 1 x 2 + x 1 x 1 x 2 x 1 + x 1 x 1 x 1 x 2 + x 1 x 1 x 2 x 1 + x 1 x 2 x 1 x 1. If the monomial basis of NSym is expressed as a sum of words of non-commutative monomials as in equation (1), we can define m A m B = αa,b m, (2) where { αa,b = # S ( [ A + B ] A ) st( S ) = A and st( S c) = B}. This shuffle product is commutative and associative and corresponds exactly to what happens when one consider the shuffle of monomials. onsidering the pairing defined on the bases of N Sym by h A, m B = δ A,B, we have the following theorem. Theorem 1 Let f, g, h NSym. Then (f), g h = f, g Proof: Since h A h B, m m D = h A, m h B, m D, then (h A ), m B m = h. S [ A ] h st(a S ) h st(a Sc ), m B m = D,E = D,E α A D,E h D h E, m B m = α A B,. α A D,Eh D h E, m B m
660 Anouk Bergeron-Brlek On the other hand, h A, m B m = h A, D αd B, m D = D αd B, h A, m D = αb, A. so the theorem follows. { orollary 4 Let βa,b = # S ( ) } [l(a)+l(b)] st( S ) = A and st( S c) = B. Then l(a) h A h B = β A,B h. Proof: Follows from Theorem 1 and the fact that (m ) = m st(s ) m st(sc ) = βa,b m A m B. S [l()] A,B onjecture 1 Let βa,b be the coefficients defined in orollary 4. Then p A p B = β A,B p. Given the previous conjecture, the following proposition follows from a simple calculation. Proposition 3 Let f, g NSym. Then ω(f g) = ω(f) ω(g). Proof: Since ω(p ) = ( 1) p (see Table 6), then ( ) ω(p A p B ) = ω βa,b p = β A,B ω(p ) = β A,B ( 1) p. Note that when l(a) + l(b) = l() then ( 1) A ( 1) B = ( 1). In the definition of the shuffle product on the p basis, we certainly have l(a) + l(b) = l(), so ω(p A p B ) = β A,B ( 1) A ( 1) B p = ( 1) A ( 1) B p A p B = ω(p A ) ω(p B ) and the proposition follows. The next corollary follows naturally from this proposition. orollary 5 Let βa,b be the coefficients defined in orollary 4. Then e A e B = β A,B e. Proof: e A e B = ω(h A ) ω(h B ) = ω(h A h B ) = ω( β A,B h ) = β A,B e.
Non-ommutative Invariants of S n and B n 661 5 Results on the invariants of the hyperoctahedral group In some preliminary work, we also consider the analogue of type B n for Q X n Sn. We consider the algebra Q X n Bn = d 0 Q X n Bn d of polynomials in Q X n which are invariant under the action of the hyperoctahedral group B n of signed permutations of [n]. The action of a signed permutation σ B n on Q X n is characterized by sending a variable x i to ±x σ(i), where σ(i) is the absolute value of σ(i). One can show that a monomial basis of Q X n Bn is given by {m A [X n ]} l(a) n, A i even where A = {A 1, A 2,..., A l(a) } is a set partition and m A [X n ] is defined as in equation (1) with a finite number of variables. In other words, a monomial basis is indexed by the set partitions with at most n parts of even cardinality. In (9), it has been proved that a basis for the centralizer algebra End Bn (V d ) of B n is also indexed by the set partitions of 2k with at most n parts of even cardinality. We have a correspondence between the centralizer algebra and the invariants of B n since End Bn (V k ) (V 2k ) Bn Q X n Bn 2k. Lemma 2 Let α d,k be the number of set partitions of d of length k with even parts. Then F k (q) = d 0 α d,k q d = 1 3... (2k 1)q 2k (1 q 2 ) (1 4 q 2 ) (1 k 2 q 2 ). Proof: We have F k (q) = F k 1 (q) (2k 1)q2 1 k 2 q, so 2 F k (q) = k 2 q 2 F k (q) + q 2 F k 1 (q) + 2(k 1)q 2 F k 1 (q). Taking the coefficient of q n on both sides yields the following recurrence: α d,k = k 2 α d 2,k + α d 2,k 1 + 2(k 1)α d 2,k 1. We would like to show that the number of set partitions of d of length k with even parts satisfies this same recurrence. Let us first denote by d,k the set of set partitions of d of length k with even parts. For a set partition A = {A 1, A 2,..., A k }, denote by m i and m i\{d} the greatest value of respectively A i and A i \{d}. Order the parts of A with the rule A i < A j if and only if m i < m j. For A d,k, suppose that d 1 A i and d A j. Let E 1 = {A d,k i = j, A i = A j > 2}, E 2 = {A d,k i < j, A i 2, A j > 2}, E 3 = {A d,k i > j, A i > 2, A j 2}, E 4 = {A d,k i = j, A i = A j = 2}, E 5 = {A d,k i < j, A i 2, A j = 2}, E 6 = {A d,k i > j, A i = 2, A j 2}. These sets are clearly mutually disjoint, and d,k = 6 i=1 E i.
662 Anouk Bergeron-Brlek Define an injection f : d,k k 2 d 2,k d 2,k 1 2(k 1) d 2,k 1 by {A 1, A 2,..., A i \{d 1, d},..., A k } if A E 1, {A 1, A 2,..., A i \{d 1} {m j\{d} },..., A j \{d, m j\{d} },..., A k } if A E 2 E 5, f(a) = {A 1, A 2,..., A j \{d} {m i\{d 1} },..., A i \{d 1, m i\{d 1} },..., A k } if A E 3 E 6, A\{{d 1, d}} if A E 4. We have that f sends E 1 E 2 E 3 to k 2 copies of d 2,k, E 4 to one copie of d 2,k 1 and E 5 E 6 to 2(k 1) copies of d 2,k 1. Denote by (i,j) d 2,k the (i, j)-th copy of d 2,k and by (i) d 2,k 1 the i-th copy of d 2,k 1. The inverse of f is defined as follows. {A 1, A 2,..., A i {d 1, d},..., A k } if A (i,i) d 2,k, {A 1, A 2,..., A i {d 1}\{m i },..., A j {d, m i },..., A k } if A (i,j) d 2,k, f 1 {A 1, A 2,..., A j {d}\{m j },..., A i {d 1, m j },..., A k } if A (j,i) d 2,k (A) =, {A 1, A 2,..., A k 1 } {{d 1, d}} if A (1) d 2,k 1, {A 1, A 2,..., A i {d 1}\{m i },..., A k 1, {d, m i }} if A (i) d 2,k 1, {A 1, A 2,..., A j {d}\{m j },..., A k 1, {d 1, m j }} if A (j) d 2,k 1. So d,k = α d,k because they satisfy the same recurrence. From the previous lemma, and the fact that a basis of Q X n Bn most n parts of even length, we get the following corollary. orollary 6 The Poincare series for the algebra Q X n Bn is is indexed by the set partitions with at d 0 dim(q X n Bn d )qd = 1 B n 1 T r(σ) q = 1 + n σ B n 1 k=1 1 3... (2k 1)q 2k (1 q 2 ) (1 4 q 2 ) (1 k 2 q 2 ). Proof: The first equality is a non-commutative analogue of Molien s Theorem, that has been proved by Dicks and Formanek (4). 6 Appendix The next two tables summarize the changes of bases and operations on NSym bases. They are followed by some examples of the coproduct and shuffle product.
Non-ommutative Invariants of S n and B n 663 e A h A m A e h p ( 1) B λ(b, A)!h B µ(0, B)p B ( 1) B λ(b, A)!e B B A µ(a, B) µ(, B)e µ(0, B) B B A p A 1 µ(0, A) B A B A B A µ(b, A)e B 1 µ(0, A) µ(a, B) µ(, B)h µ(0, B) B µ(b, A)h B B A B A µ(0, B) p B B A µ(a, B)p B B A Tab. 1: hanges of bases (see (10)). For e A, h A, p A in terms of m, see section 3. f f A f B (f A ) ω(f A ) f A f B e e st(a S ) e st(a S c ) e A B h β A A,B e h m p Proposition 1 h A B orollary 2 ([n] [k])=a B (2) Proposition 3.2 p A B (1) Lemma 4.1 m S [n] Proposition 2 S [n] orollary 3 (2) (6) (1) Remark 4.4 h st(a S ) h st(a S c ) m st(as ) m st(as c ) p st(as ) p st(as c ) (10) orollary 5 e A (10) orollary 4? ( 1) A p A (10) Theorem 3.5 Tab. 2: Operations on NSym bases, where A [n] and B [k]. βa,b h αa,b m Equation 2 βa,b p onjecture 1 f (f {124,3} ) 1 h h {124,3} + h {1} h {12,3} + 2 h {1} h {13,2} + h {1} h {123} + 3 h {12} h {1,2} +3 h {1,2} h {12} + h {123} h {1} + 2 h {13,2} h {1} + h {12,3} h {1} + h {124,3} 1 m 1 m {124,3} + m {1} m {123} + m {123} m {1} + m {124,3} 1 f f {12} f {1,2} h h {1,2,34} + h {13,2,4} + h {1,23,4} + h {14,2,3} + h {1,24,3} + h {12,3,4} m m {12,3,4} + m {13,2,4} + m {14,2,3} + m {1,23,4} + m {1,24,3} + m {1,2,34} +3 m {134,2} + 3 m {123,4} + 3 m {1,234} + 3 m {124,3} Tab. 3: Examples of coproduct and shuffle product.
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