INEQUALITIES BETWEEN LITTLEWOOD RICHARDSON COEFFICIENTS

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INEQUALITIES BETWEEN LITTLEWOOD-RICHARDSON COEFFICIENTS FRANC¸OIS BERGERON, RICCARDO BIAGIOLI, AND MERCEDES H. ROSAS Abstract. We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes. Contents 1. Introduction 1 2. Combinatorial properties of the ?-operation and implications 3 3. Main results 8 4. Proofs of the combinatorial properties 9 5. Extension of the ?-operation to tableaux 13 6. Background on Littlewood-Richardson coefficients 15 7. Proof of special instances 16 8. Reduction to a finite set of pairs in bounded height case 21 9. Final remarks 23 10. Acknowledgments 23 References 23 1. Introduction In the course of their study of Horn type inequalities for eigenvalues and singular values of complex matrices, Fomin, Fulton, Li, and Poon [2] come up with a very interesting con- jecture concerning the Schur-positivity of special differences of products of Schur functions. More precisely, they consider differences of the form sµ?s?? ? sµs?, where µ? and ?? are partitions constructed from an ordered pair of partitions µ and ? through a seemingly strange procedure at first glance.

coefficients c?µ ?

littlewood-richardson coefficient

µk ?

sb sb

partition

schur functions

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INEQUALITIES BETWEEN LITTLEWOOD-RICHARDSON COEFFICIENTS FRAN¸COISBERGERON,RICCARDOBIAGIOLI,ANDMERCEDESH.ROSAS Abstract. We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many inﬁnite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a ﬁnite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.

Contents 1. Introduction 2. Combinatorial properties of the ∗ -operation and implications 3. Main results 4. Proofs of the combinatorial properties 5. Extension of the ∗ -operation to tableaux 6. Background on Littlewood-Richardson coeﬃcients 7. Proof of special instances 8. Reduction to a ﬁnite set of pairs in bounded height case 9. Final remarks 10. Acknowledgments References

1 3 8 9 13 15 16 21 23 23 23

1. Introduction In the course of their study of Horn type inequalities for eigenvalues and singular values of complex matrices, Fomin, Fulton, Li, and Poon [2] come up with a very interesting con-jecture concerning the Schur-positivity of special diﬀerences of products of Schur functions. More precisely, they consider diﬀerences of the form s  ∗ s ν ∗ − s  s ν  where  ∗ and ν ∗ are partitions constructed from an ordered pair of partitions  and ν through a seemingly strange procedure at ﬁrst glance. In our presentation, their transfor-mation (  ν ) 7→ (  ∗  ν ∗ ) on ordered pairs of partitions, will rather be denoted (1.1) (  ν ) 7−→ (  ν ) ∗ = ( λ (  ν )  ρ (  ν )) F. Bergeron is supported in part by NSERC and FQRNT. 1

2FRAN¸COISBERGERON,RICCARDOBIAGIOLI,ANDMERCEDESH.ROSAS and will be called the ∗ -operation. As we shall see, this change of notation is essential in order to simplify the presentation of the many nice combinatorial properties of this ∗ operation. On the other hand, it underlines that both entries, λ and ρ of the image (  ν ) of (  ν ), actually depend on both  and ν . With this slight change of notation, the original deﬁnition of the ∗ -operation is as follows. Let  = (  1   2       n ) and ν = ( ν 1  ν 2      ν n ) two partitions with the same number of parts, allowing zero parts. From these, two new partitions λ (  ν ) = ( λ 1  λ 2      λ n ) and ρ (  ν ) = ( ρ 1  ρ 2      ρ n ) are constructed as follows (1.2) λρ jk ::== ν jk −− jk ++1#+ { j # |{ 1 k |≤ 1 j ≤≤ kn ≤ νn j  − j k ≥− k k > − νk j }− ; j }  Although this deﬁnition does not make it immediately clear, both λ (  ν ) and ρ (  ν ) are truly partitions, and they are such that | λ (  ν ) | + | ρ (  ν ) | = |  | + | ν |  where as usual |  | denotes the sum of the parts of  . Recall that the product of two Schur functions can always be expanded as a linear com-bination s  s ν = X c θν s θ  θ of Schur functions indexed by partitions θ of the integer n = |  | + | ν | , since these Schur functions constitute a linear basis of the homogeneous symmetric functions of degree n . It is a particularly nice feature of this expansion that the coeﬃcients c θν are always non-negative integers. They are called the Littlewood-Richardson coeﬃcients . More generally, we say that a symmetric function is Schur positive whenever the coeﬃcients in its expansion, in the Schur function basis, are all non-negative integers. For more details on symmetric function theory see Macdonald’s classical book [3], whose notations we will mostly follow. We can then state the following: Conjecture 1.1 (Fomin-Fulton-Li-Poon) . For any pair of partitions (  ν ) , if (  ν ) ∗ = ( λ ρ )  then the symmetric function (1.3) s λ s ρ − s  s ν is Schur-positive. In other words, this says that c θν ≤ c λθρ , for all θ such that s θ appears in the expansion of s  s ν . For an example of one of the simplest case of the ∗ -operation, let  = ( a ) and ν = ( b ), with a > b , be two one-part partitions. In this case, we get (( a )  ( b )) ∗ = ( a − 1  b + 1) 

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