STAIRCASE SKEW SCHUR FUNCTIONS ARE SCHUR P-POSITIVE 1 ...
STAIRCASE SKEW SCHUR FUNCTIONS ARE SCHUR P -POSITIVE FEDERICO ARDILA AND LUIS G. SERRANO
Abstract. We prove Stanley’s conjecture that, if δn is the staircase shape, then the skew Schur functions sδn /µ are non-negative sums of Schur P -functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function sδn /δn−2 , we discuss connections with Eulerian numbers and alternating permutations.
1. introduction The Schur functions sλ , indexed by partitions λ, form a basis for the ring Λ of symmetric functions. These are very important objects in algebraic combinatorics. They play a fundamental role in the study of the representations of the symmetric group and the general linear group, and the cohomology ring of the Grassmannian [4]. The Schur P -functions Pλ , indexed by strict partitions, form a basis for an important subring Γ of Λ. They are crucial in the study of the projective representations of the symmetric group, and the cohomology ring of the isotropic Grassmannian [10] [11]. The goal of this paper is to prove the following conjecture of Richard Stanley [15]: If δn is the staircase shape and µ ⊂ δn , then the staircase skew Schur function sδn /µ , which belongs to the ring Γ, is a nonnegative sum of Schur P -functions. We find a combinatorial interpretation for the coefficients in this expansion in terms of Shimozono’s compatible fillings [13]. Furthermore, we discuss connections between the special case of the skew Schur function sδn /δn−2 and alternating permutations, and show an expansion of these in terms of the elementary symmetric functions. The paper is organized as follows. In Section 2 we recall some basic definitions, including Schur and Schur P -functions. In Section 3 we discuss the staircase Schur functions and prove that they are indeed in the subring Γ of Λ generated by the Schur P -functions. In Section 4 we state our main result, Theorem 4.10, which states that the (non-negative integer) coefficients of Pλ is the number of “δn /µcompatible” fillings of the shifted shape λ. In Section 5 we prove the key proposition that, in the particular case of staircase skew shapes δn /µ, jeu de taquin respects δn /µ–compatibility. Finally in Section 6 we prove Theorem 4.10. Date: August 16, 2011. 2000 Mathematics Subject Classification. Primary 05E05; Secondary 05A05, 05E10, 20C25, 20C30. Key words and phrases. Schur functions, Schur P -functions, shifted tableaux, Eulerian numbers, alternating permutations. The first author was partially supported by the National Science Foundation CAREER Award DMS-0956178, the National Science Foundation Grant DMS-0801075, and the SFSU-Colombia Combinatorics Initiative. The second author was supported by a National Science and Engineering Council of Canada (NSERC) PDF Award. 1
2
FEDERICO ARDILA AND LUIS G. SERRANO
The Schur P -positivity of staircase Schur functions has also been proved independently by Elizabeth Dewitt and will appear in her forthcoming thesis [2] Acknowledgments. We would like to thank Richard Stanley for telling us about his conjecture and about Proposition 3.4. [15] We also thank Ira Gessel, Peter Hoffman, Tadeusz J´ ozefiak, Bruce Sagan, and John Stembridge for valuable conversations. 2. Preliminaries A partition is a sequence λ = (λ1 , λ2 , . . . , λl ) ∈ Zl with λ1 ≥ λ2 ≥ · · · ≥ λl > 0. The Ferrers diagram, or shape of λ is an array of square cells in which the i-th row has λi cells, and is left justified with respect to the top row. The size of λ is |λ| := λ1 + λ2 + · · · + λl . We denote the number of rows of λ by `(λ) := l. A strict partition is a sequence λ = (λ1 , λ2 , . . . , λl ) ∈ Zl such that λ1 > λ2 > · · · > λl > 0. The shifted diagram, or shifted shape of λ is an array of square cells in which the i-th row has λi cells, and is shifted i − 1 units to the right with respect to the top row. For example, the shape (5, 3, 2) and the shifted shape (5, 3, 2), of size 10 and length 3, are shown below.
A skew (shifted) diagram (or shape) λ/µ is obtained by removing a (shifted) shape µ from a larger shape λ containing µ. A semistandard Young tableau or SSYT T of shape λ is a filling of a Ferrers shape λ with letters from the alphabet X = {1 < 2 < · · · } which is weakly increasing along the rows and strictly increasing down the columns. A shifted semistandard Young tableau or shifted SSYT T of shape λ is a filling of a shifted shape λ with letters from the alphabet X 0 = {10 < 1 < 20 < 2 < · · · } such that: • rows and columns of T are weakly increasing; • each k appears at most once in every column; • each k 0 appears at most once in every row; • there are no primed entries on the main diagonal. If T is a filling of a shape λ, we write sh(T ) := λ. The content of a (shifted) SSYT T is the vector (a1 , a2 , . . .), where ai is the number of times the letters i and i0 appear in T . A (shifted) SSYT is standard, if it contains the letters 1, 2, . . . , |λ|, each exactly once. In the shifted case, these letters are all unprimed. If that is the case, we call it a (shifted) SYT. A skew (shifted) Young tableau is defined analogously. Example 2.1. The following are examples of a SSYT and a shifted SSYT, both having shape λ = (5, 3, 2) and content (2, 1, 1, 2, 2, 1, 0, 0, 1). 1 1 2 3 5 4 4 5 6 9
1 1 2 30 5 4 4 5 6 90
In a SYT or a shifted SYT T , the pair of entries (i, j), where i < j, forms an ascent if j is located weakly north and strictly east of i. We abbreviate and say
STAIRCASE SKEW SCHUR FUNCTIONS ARE SCHUR P -POSITIVE
3
that j is northEast of i. The pair (i, j) forms a descent if j is located strictly south and weakly west, or Southwest, of i. Note that (i, j) could be neither an ascent nor a descent. When j = i + 1, the pair (i, i + 1) must be either an ascent or a descent, and we abbreviate and call i an ascent or a descent as appropriate. An entry i forms a peak if i − 1 is an ascent and i is a descent. Example 2.2. The figure below shows a SYT of shape δ4 := (4, 3, 2, 1) and a shifted SYT of shape (5, 3, 2), both with descent set (2, 4, 5, 7, 9), ascent set (1, 3, 6, 8), and peak set (2, 4, 7, 9). 1 2 4 7 1 2 4 7 9 3 5 9 3 5 8 6 10 6 10 8 For a (shifted) Young tableau T with content (a1 , a2 , . . .), we let xT = xa1 1 xa2 2 · · · . For each partition λ, the Schur function sλ is defined as the generating function for semistandard Young tableaux of shape λ, namely X sλ = sλ (x1 , x2 , . . .) := xT . sh(T )=λ
It is well known (see e.g., [14]) that the power sum symmetric functions pi = pi (x1 , x2 , . . .) := xi1 + xi2 + · · · are a generating set, and the Schur functions sλ are a linear basis, for the ring Λ of symmetric functions. For each strict partition λ, the Schur P -function Pλ is defined as the generating function for shifted Young tableaux of shape λ, namely X Pλ = Pλ (x1 , x2 , . . .) := xT . sh(T )=λ
The Schur P -functions form a basis for the subring Γ of Λ generated by the odd power sums, Γ := Q[p1 , p3 , . . .]. This ring also has the presentation Γ = {f ∈ Λ : f (t, −t, x1 , x2 , . . .) = f (x1 , x2 , . . .)}. See, e.g., [10]. The skew Schur functions sλ/µ and the skew Schur P -functions Pλ/µ are defined similarly for a skew (shifted) shape λ/µ. 3. The skew Schur functions sδn /δn−2 and sδn /µ . Definition 3.1. The staircase δn is the Pshape (n, n−1, . . . , 2, 1). Denote sδn /δn−2 =: F2n−1 , and let F = F(x1 , x2 , . . .) := n≥1 F2n−1 . The symmetric function F2n−1 is one of the main subjects of study of this paper. It has nice expansions in terms of the power and elementary symmetric functions. Definition 3.2. A permutation a1 a2 . . . an of {1, . . . , n} is said to be alternating if a1 < a2 > a3 < a4 > · · · . Proposition 3.3. ([3]) Let Ek be the number of alternating permutations of {1, . . . , k}, Q mi and let zλ := i≥1 imi ! for the partition λ = 1m1 2m2 · · · . We have X El(λ) F2n−1 = pλ zλ λ∈OP (2n−1)
where OP (2n − 1) is the set of partitions of 2n − 1 into odd parts.
4
FEDERICO ARDILA AND LUIS G. SERRANO
The following proposition expresses the F in terms of the elementary symmetric functions. Equivalent formulas appear in [1], [5], [6, p. 9] and [7, Corollary 4.2.20]. Proposition 3.4. We have F= where ek =
P
i1 2 > 1.
12
FEDERICO ARDILA AND LUIS G. SERRANO
Theorem 6.2 (Stembridge [16]). Fix a shifted SYT U of shape λ. We have X Pλ = gµλ sµ , µ
where gµλ is the number of SYT T of shape µ such that jdt(T ) = U . We have now assembled all the ingredients to prove the main theorem. Proof of Theorem 4.10. Denote the set of shifted standard Young tableaux of shape δn /µ by ShSYT, and the set of (shifted) standard δn /µ–compatible tableaux by CompSYT(CompShSYT). By Theorem 6.1 we have X sδn /µ = ssh(T ) T ∈CompSYT
=
X
X
ssh(T ) .
U ∈ShSYT T ∈CompSYT : U =jdt(T )
By Proposition 5.4 and Theorem 6.2 respectively, this equals X X sδn /µ = ssh(T ) U ∈CompShSYT T ∈SYT : U =jdt(T )
=
X
Psh(U )
U ∈CompShSYT
as we wished to prove.
7. Further Work
• As mentioned earlier, the Schur and the Schur P -functions are related to the representations and the projective representations of the symmetric group, and to the cohomology of the Grassmannian and the isotropic Grassmannian. The representation theoretic and geometric significance of Theorem 4.10 should be explored. • Theorem 4.10 implies that if λ/µ is a disjoint union of staircase skew shapes and their 180 degree rotations, then sλ/µ is Schur P -positive. It is natural to wonder whether these are the only skew Schur functions which are Schur P -positive. In fact, Dewitt [2] has proved the stronger statement that these are the only skew Schur functions which are linear combinations of Schur P -functions. References [1] L. Carlitz, Enumeration of up-down sequences, Discrete Math 4 (1973), 273–286. [2] E. Dewitt, Identities Relating Schur s-Functions and Q-Functions, Ph.D. Thesis, University of Michigan, 2012. [3] H. O. Foulkes, Enumeration of permutations with prescribed up-down and inversion sequences, Discrete Math 15 (1976), 235–252. [4] W. Fulton. Young tableaux. London Mathematical Society Student Texts 35. Cambridge University Press, 1997. [5] I. M. Gessel, Generating Functions and Enumeration of Sequences, Ph.D. Thesis, Massachusetts Institute of Technology, 1977. [6] I. M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), 257–285. [7] I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley & Sons, New York, 1983 (Dover Reprint, 2004).
STAIRCASE SKEW SCHUR FUNCTIONS ARE SCHUR P -POSITIVE
13
[8] M. D. Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Math 99 (1992), 79–113. [9] M. D. Haiman, On mixed insertion, symmetry, and shifted Young tableaux, J. Combin. Theory Ser. A 50 (1989), 196–225. [10] P. N. Hoffman and J. F. Humphreys, Projective representations of the symmetric groups, Oxford University Press, 1992. [11] T. J´ ozefiak, Schur Q-functions and cohomology of isotropic Grassmannians. Math. Proc. Camb. Phil. Soc. 109 (1991), 471–478. [12] B. Sagan, Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley, J. Combin. Theory Ser. A 45 (1987), 62–103. [13] M. Shimozono, Multiplying Schur Q-functions, J. Combin. Theory Ser. A 87 (1999), no. 1, 198–232. [14] R. P. Stanley, Enumerative combinatorics, V2, Cambridge University Press, 1999. [15] R. P. Stanley, personal communication, 2001. [16] J. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), 87–134. Department of Mathematics, San Francisco State University. E-mail address: [email protected] URL: http://math.sfsu.edu/federico/ ´ du Que ´bec a ` Montre ´al. LaCIM, Universite E-mail address: [email protected] URL: http://www.thales.math.uqam.ca/~serrano/
Abstract. We prove Stanley’s conjecture that, if δn is the staircase shape, then the skew Schur functions sδn /µ are non-negative sums of Schur P -functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function sδn /δn−2 , we discuss connections with Eulerian numbers and alternating permutations.
1. introduction The Schur functions sλ , indexed by partitions λ, form a basis for the ring Λ of symmetric functions. These are very important objects in algebraic combinatorics. They play a fundamental role in the study of the representations of the symmetric group and the general linear group, and the cohomology ring of the Grassmannian [4]. The Schur P -functions Pλ , indexed by strict partitions, form a basis for an important subring Γ of Λ. They are crucial in the study of the projective representations of the symmetric group, and the cohomology ring of the isotropic Grassmannian [10] [11]. The goal of this paper is to prove the following conjecture of Richard Stanley [15]: If δn is the staircase shape and µ ⊂ δn , then the staircase skew Schur function sδn /µ , which belongs to the ring Γ, is a nonnegative sum of Schur P -functions. We find a combinatorial interpretation for the coefficients in this expansion in terms of Shimozono’s compatible fillings [13]. Furthermore, we discuss connections between the special case of the skew Schur function sδn /δn−2 and alternating permutations, and show an expansion of these in terms of the elementary symmetric functions. The paper is organized as follows. In Section 2 we recall some basic definitions, including Schur and Schur P -functions. In Section 3 we discuss the staircase Schur functions and prove that they are indeed in the subring Γ of Λ generated by the Schur P -functions. In Section 4 we state our main result, Theorem 4.10, which states that the (non-negative integer) coefficients of Pλ is the number of “δn /µcompatible” fillings of the shifted shape λ. In Section 5 we prove the key proposition that, in the particular case of staircase skew shapes δn /µ, jeu de taquin respects δn /µ–compatibility. Finally in Section 6 we prove Theorem 4.10. Date: August 16, 2011. 2000 Mathematics Subject Classification. Primary 05E05; Secondary 05A05, 05E10, 20C25, 20C30. Key words and phrases. Schur functions, Schur P -functions, shifted tableaux, Eulerian numbers, alternating permutations. The first author was partially supported by the National Science Foundation CAREER Award DMS-0956178, the National Science Foundation Grant DMS-0801075, and the SFSU-Colombia Combinatorics Initiative. The second author was supported by a National Science and Engineering Council of Canada (NSERC) PDF Award. 1
2
FEDERICO ARDILA AND LUIS G. SERRANO
The Schur P -positivity of staircase Schur functions has also been proved independently by Elizabeth Dewitt and will appear in her forthcoming thesis [2] Acknowledgments. We would like to thank Richard Stanley for telling us about his conjecture and about Proposition 3.4. [15] We also thank Ira Gessel, Peter Hoffman, Tadeusz J´ ozefiak, Bruce Sagan, and John Stembridge for valuable conversations. 2. Preliminaries A partition is a sequence λ = (λ1 , λ2 , . . . , λl ) ∈ Zl with λ1 ≥ λ2 ≥ · · · ≥ λl > 0. The Ferrers diagram, or shape of λ is an array of square cells in which the i-th row has λi cells, and is left justified with respect to the top row. The size of λ is |λ| := λ1 + λ2 + · · · + λl . We denote the number of rows of λ by `(λ) := l. A strict partition is a sequence λ = (λ1 , λ2 , . . . , λl ) ∈ Zl such that λ1 > λ2 > · · · > λl > 0. The shifted diagram, or shifted shape of λ is an array of square cells in which the i-th row has λi cells, and is shifted i − 1 units to the right with respect to the top row. For example, the shape (5, 3, 2) and the shifted shape (5, 3, 2), of size 10 and length 3, are shown below.
A skew (shifted) diagram (or shape) λ/µ is obtained by removing a (shifted) shape µ from a larger shape λ containing µ. A semistandard Young tableau or SSYT T of shape λ is a filling of a Ferrers shape λ with letters from the alphabet X = {1 < 2 < · · · } which is weakly increasing along the rows and strictly increasing down the columns. A shifted semistandard Young tableau or shifted SSYT T of shape λ is a filling of a shifted shape λ with letters from the alphabet X 0 = {10 < 1 < 20 < 2 < · · · } such that: • rows and columns of T are weakly increasing; • each k appears at most once in every column; • each k 0 appears at most once in every row; • there are no primed entries on the main diagonal. If T is a filling of a shape λ, we write sh(T ) := λ. The content of a (shifted) SSYT T is the vector (a1 , a2 , . . .), where ai is the number of times the letters i and i0 appear in T . A (shifted) SSYT is standard, if it contains the letters 1, 2, . . . , |λ|, each exactly once. In the shifted case, these letters are all unprimed. If that is the case, we call it a (shifted) SYT. A skew (shifted) Young tableau is defined analogously. Example 2.1. The following are examples of a SSYT and a shifted SSYT, both having shape λ = (5, 3, 2) and content (2, 1, 1, 2, 2, 1, 0, 0, 1). 1 1 2 3 5 4 4 5 6 9
1 1 2 30 5 4 4 5 6 90
In a SYT or a shifted SYT T , the pair of entries (i, j), where i < j, forms an ascent if j is located weakly north and strictly east of i. We abbreviate and say
STAIRCASE SKEW SCHUR FUNCTIONS ARE SCHUR P -POSITIVE
3
that j is northEast of i. The pair (i, j) forms a descent if j is located strictly south and weakly west, or Southwest, of i. Note that (i, j) could be neither an ascent nor a descent. When j = i + 1, the pair (i, i + 1) must be either an ascent or a descent, and we abbreviate and call i an ascent or a descent as appropriate. An entry i forms a peak if i − 1 is an ascent and i is a descent. Example 2.2. The figure below shows a SYT of shape δ4 := (4, 3, 2, 1) and a shifted SYT of shape (5, 3, 2), both with descent set (2, 4, 5, 7, 9), ascent set (1, 3, 6, 8), and peak set (2, 4, 7, 9). 1 2 4 7 1 2 4 7 9 3 5 9 3 5 8 6 10 6 10 8 For a (shifted) Young tableau T with content (a1 , a2 , . . .), we let xT = xa1 1 xa2 2 · · · . For each partition λ, the Schur function sλ is defined as the generating function for semistandard Young tableaux of shape λ, namely X sλ = sλ (x1 , x2 , . . .) := xT . sh(T )=λ
It is well known (see e.g., [14]) that the power sum symmetric functions pi = pi (x1 , x2 , . . .) := xi1 + xi2 + · · · are a generating set, and the Schur functions sλ are a linear basis, for the ring Λ of symmetric functions. For each strict partition λ, the Schur P -function Pλ is defined as the generating function for shifted Young tableaux of shape λ, namely X Pλ = Pλ (x1 , x2 , . . .) := xT . sh(T )=λ
The Schur P -functions form a basis for the subring Γ of Λ generated by the odd power sums, Γ := Q[p1 , p3 , . . .]. This ring also has the presentation Γ = {f ∈ Λ : f (t, −t, x1 , x2 , . . .) = f (x1 , x2 , . . .)}. See, e.g., [10]. The skew Schur functions sλ/µ and the skew Schur P -functions Pλ/µ are defined similarly for a skew (shifted) shape λ/µ. 3. The skew Schur functions sδn /δn−2 and sδn /µ . Definition 3.1. The staircase δn is the Pshape (n, n−1, . . . , 2, 1). Denote sδn /δn−2 =: F2n−1 , and let F = F(x1 , x2 , . . .) := n≥1 F2n−1 . The symmetric function F2n−1 is one of the main subjects of study of this paper. It has nice expansions in terms of the power and elementary symmetric functions. Definition 3.2. A permutation a1 a2 . . . an of {1, . . . , n} is said to be alternating if a1 < a2 > a3 < a4 > · · · . Proposition 3.3. ([3]) Let Ek be the number of alternating permutations of {1, . . . , k}, Q mi and let zλ := i≥1 imi ! for the partition λ = 1m1 2m2 · · · . We have X El(λ) F2n−1 = pλ zλ λ∈OP (2n−1)
where OP (2n − 1) is the set of partitions of 2n − 1 into odd parts.
4
FEDERICO ARDILA AND LUIS G. SERRANO
The following proposition expresses the F in terms of the elementary symmetric functions. Equivalent formulas appear in [1], [5], [6, p. 9] and [7, Corollary 4.2.20]. Proposition 3.4. We have F= where ek =
P
i1 2 > 1.
12
FEDERICO ARDILA AND LUIS G. SERRANO
Theorem 6.2 (Stembridge [16]). Fix a shifted SYT U of shape λ. We have X Pλ = gµλ sµ , µ
where gµλ is the number of SYT T of shape µ such that jdt(T ) = U . We have now assembled all the ingredients to prove the main theorem. Proof of Theorem 4.10. Denote the set of shifted standard Young tableaux of shape δn /µ by ShSYT, and the set of (shifted) standard δn /µ–compatible tableaux by CompSYT(CompShSYT). By Theorem 6.1 we have X sδn /µ = ssh(T ) T ∈CompSYT
=
X
X
ssh(T ) .
U ∈ShSYT T ∈CompSYT : U =jdt(T )
By Proposition 5.4 and Theorem 6.2 respectively, this equals X X sδn /µ = ssh(T ) U ∈CompShSYT T ∈SYT : U =jdt(T )
=
X
Psh(U )
U ∈CompShSYT
as we wished to prove.
7. Further Work
• As mentioned earlier, the Schur and the Schur P -functions are related to the representations and the projective representations of the symmetric group, and to the cohomology of the Grassmannian and the isotropic Grassmannian. The representation theoretic and geometric significance of Theorem 4.10 should be explored. • Theorem 4.10 implies that if λ/µ is a disjoint union of staircase skew shapes and their 180 degree rotations, then sλ/µ is Schur P -positive. It is natural to wonder whether these are the only skew Schur functions which are Schur P -positive. In fact, Dewitt [2] has proved the stronger statement that these are the only skew Schur functions which are linear combinations of Schur P -functions. References [1] L. Carlitz, Enumeration of up-down sequences, Discrete Math 4 (1973), 273–286. [2] E. Dewitt, Identities Relating Schur s-Functions and Q-Functions, Ph.D. Thesis, University of Michigan, 2012. [3] H. O. Foulkes, Enumeration of permutations with prescribed up-down and inversion sequences, Discrete Math 15 (1976), 235–252. [4] W. Fulton. Young tableaux. London Mathematical Society Student Texts 35. Cambridge University Press, 1997. [5] I. M. Gessel, Generating Functions and Enumeration of Sequences, Ph.D. Thesis, Massachusetts Institute of Technology, 1977. [6] I. M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), 257–285. [7] I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley & Sons, New York, 1983 (Dover Reprint, 2004).
STAIRCASE SKEW SCHUR FUNCTIONS ARE SCHUR P -POSITIVE
13
[8] M. D. Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Math 99 (1992), 79–113. [9] M. D. Haiman, On mixed insertion, symmetry, and shifted Young tableaux, J. Combin. Theory Ser. A 50 (1989), 196–225. [10] P. N. Hoffman and J. F. Humphreys, Projective representations of the symmetric groups, Oxford University Press, 1992. [11] T. J´ ozefiak, Schur Q-functions and cohomology of isotropic Grassmannians. Math. Proc. Camb. Phil. Soc. 109 (1991), 471–478. [12] B. Sagan, Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley, J. Combin. Theory Ser. A 45 (1987), 62–103. [13] M. Shimozono, Multiplying Schur Q-functions, J. Combin. Theory Ser. A 87 (1999), no. 1, 198–232. [14] R. P. Stanley, Enumerative combinatorics, V2, Cambridge University Press, 1999. [15] R. P. Stanley, personal communication, 2001. [16] J. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), 87–134. Department of Mathematics, San Francisco State University. E-mail address: [email protected] URL: http://math.sfsu.edu/federico/ ´ du Que ´bec a ` Montre ´al. LaCIM, Universite E-mail address: [email protected] URL: http://www.thales.math.uqam.ca/~serrano/
