aa r X i v : . [ m a t h . C O ] S e p THE FIRST TERM OF PLETHYSMS
K. IIJIMA
Abstract.
Plethysm of two Schur functions can be expressed asa linear combination of Schur functions, and monomial symmetricfunctions. In this paper, we express the coefficients combinatoriallyin the case of monomial symmetric functions. And by using it, wedetermine the first term of the plethysm with respect to Schurfunctions under the reverse lexicographic order. Introduction
Let λ and µ be partitions of positive integers m and n , respectively.The plethysm s λ [ s µ ] is the symmetric function obtained by substitutingthe monomials in s µ for the variables of s λ . D.E.Littlewood introducedthis operation in 1936[9].Plethysm of two Schur functions is expressed as a linear combinationof Schur functions; s λ [ s µ ] = X ν ⊢ mn a νλ [ µ ] s ν , where ν ⊢ mn means that ν is a partition of mn .Plethysm appear in some fields, especially representation theory. Forexample, the coefficient a νλ [ µ ] is equal to the multiplicity of the irre-ducible GL N -module of highest weight ν in a certain GL N -module,and the multiplicity of the irreducible S mn -module type of ν in a cer-tain S n ≀ S m -module, where GL N , S mn , and S n ≀ S m are the general lineargroup, the symmetric group, and the wreath product of the symmetricgroup , respectively. (See [10;Chapter.1 Appendix A and AppendixB].) From these interpretations, we see that each of the coefficient a νλ [ µ ] is a nonnegative integer.One of most fundamental problems for plethysm is expressing the co-efficients a νλ [ µ ] combinatorially like Kostka coefficients and Littlewood-Richardson coefficients. In the case of λ = (2) or (1 ), Carr´ e and Leclerc[4] found a combinatorial description by using domino tableaux. How-ever it is generally open problem. Mathematics Subject Classification.
Primary 05E05.
In [2] Agaoka gave the table of a νλ [ µ ] up to mn = 16, and in [11] ananother method for calculating a νλ [ µ ] is given.Moreover plethysm is a necessary tool when we consider some geo-metric problems. For example, in [1] Agaoka found a new obstructionof local isometric imbeddings of Riemannian submanifolds with codi-mension 2 by calculating the plethysm s [ s , ].Now we explain our approach briefly. Plethysm is also expressed asa sum of monomial symmetric functions; s λ [ s µ ] = X ν ⊢ mn Y νλ [ µ ] m ν where m ν denotes the monomial symmetric function corresponding topartition ν . We give a combinatorial description of the coefficients Y νλ [ µ ] (see section 3), and using this, we determine the first term of s λ [ s µ ] (seesection 4). (The first term is the maximal element among the partition ν satisfying a νλ [ µ ] = 0 with respect to the reverse lexicographic order.)2. Preliminary
Young tableaux, symmetric functions.
We start with intro-ducing the notations. For a positive integer m , let [1 , m ] = { i ∈ Z | ≤ i ≤ m } be the interval of integers between 1 and m . For a positiveinteger n , a partition of n is a non-increasing sequences of non-negativeintegers summing to n . We write λ ⊢ n if λ is a partition of size n .And we use the same notation λ to represent the Young diagram cor-responding to λ . Let s λ , m λ , and h λ denote Schur function, monomialsymmetric function, and complete symmetric function correspondingto λ , respectively. Here we use x , x , · · · as variables. And we definea symmetric bilinear form h , i on the ring of symmetric functions asfollows; h s λ , s µ i = δ λ,µ . Next we introduce notations for Young tableaux. For a given Youngdiagram λ , a Young tableau (of shape µ ) is a map from the set of cells(in the Young diagram λ ) to a totally ordered set S . For a given Youngtableau T , the image of ( i, j ) is denoted by T ( i, j ) and called the ( i, j ) entry of T . A semi standard tableau is a Young tableau whose entriesincrease weakly along the rows and increase strictly down the columns.For a Young diagram λ , SSTab( λ, S ) denotes the set of semi standardtableaux of shape λ .In particular we can take the set of positive integers as a totally or-dered set S . In this case we write SSTab( λ, [1 , m ]) simply SSTab( λ ) ≤ m .For a Young tableau T , the weight of T is the sequence wt( T ) = HE FIRST TERM OF PLETHYSMS 3 ( µ , µ , · · · ), where µ k is the number of T ( i, j ) equal to k . We denote bySSTab( λ ; µ ) the set of semi standard tableaux of shape λ with weight µ . For a tableau T ∈ SSTab( λ ; µ ), we define x T = x wt( T ) = x µ x µ · · · .Next we define a total order in SSTab( λ ) which is used in section 3.For a given semi standard tableau T , by reading T from left to right inconsecutive rows, starting from the top to bottom, we obtain the word word( T ). We define a total order > on the set of words (in which entryis a positive integer) as the lexicographic order. Definition 2.1.
Let
T, U ∈ SSTab( λ ) . We define T > U if word( T ) > word( U ) . Example 2.2.
Let T = 1 1 23 , T = 1 1 24 , T = 1 2 22 Then word( T ) = 1123 , word( T ) = 1124 , word( T ) = 1222 . Thus T < T < T . Now we recall well-known results for Kostka coefficients.
Definition 2.3.
For λ, µ ⊢ n , the Kostka coefficient K λ,µ is defined by s λ = X µ ⊢ n K λ,µ m µ . Similarly the inverse Kostka coefficient K − λ,µ is given by m λ = X µ ⊢ n K − λ,µ s µ . By a simple consideration, we have the following;
Proposition 2.4. h λ = X µ ⊢ n K λ,µ s µ . Next theorem supply us with a combinatorially expression of Kostkacoefficients.
Theorem 2.5.
Let λ, µ ⊢ n , then we have K λ,µ = λ ; µ ) . Remark . For K − λ,µ we also have a combinatorially expression.[5]From this theorem we have some corollaries which we will use later. Corollary 2.6.
We introduce a total order on the set of Young dia-grams by the reverse lexicographic order. ( see [10]) Then for λ, µ ⊢ n , (i) If λ < µ , then K λ,µ = 0 . K. IIJIMA (ii) If λ < µ , then K − λ,µ = 0 . For a positive integer m , by putting x m +1 = x m +2 = · · · = 0 in thetheorem, we have the next corollary. Corollary 2.7.
For a Schur function with m variables s λ ( x , · · · , x m ) ,we have s λ ( x , · · · , x m ) = X T ∈ SSTab( µ ) ≤ m x T . plethysm. Let f and g be two symmetric functions and write g asa sum of monomials: g = P α ∈ N ∞ c α x α . Introduce the set of fictitiousvariables y i defined byΠ(1 + y i t ) = Π α ∈ N ∞ (1 + x α t ) c α and define f [ g ] = f ( y , y , · · · ). If f is n -th symmetric function and g is m -th, then f [ g ] is nm -th symmetric function. We call this multipleon the set of symmetric functions plethysm . Proposition 2.8.
Let f and g be two symmetric functions. We restrict g to s -variables and write it as a sum of monomials: g ( x , · · · , x s , , , · · · ) = N X i =1 x α ( i ) . Then, f [ g ]( x , · · · , x s , , , · · · ) = f ( x α (1) , · · · , x α ( N ) , , , · · · ) . That is, f [ g ]( x , · · · , x s , , , · · · ) is the symmetric polynomial obtainedby substituting monomials in f (together with multiplicity) for the vari-ables in g . The expression of plethysm in monomial symmetricfunctions
Let λ ⊢ m , µ ⊢ n and ν ⊢ mn . We put a copy the Young diagram µ in each cell of the Young diagram λ , and denote such a diagramby λ [ µ ]. For example, if λ = (3 ,
1) and µ = (3 , HE FIRST TERM OF PLETHYSMS 5
Fig.1
Definition 3.1.
A semi standard tableau of shape λ [ µ ] is a semi stan-dard tableau T : λ → SSTab( µ ) in the sense of Definition 2.1. Namelyit is filled with mn number of positive integers and it satisfies followingtwo conditions; ( i ) . Each Young tableau of shape µ is a semi standard tableau. ( ii ) . These m number of semi standard tableaux form a semi stan-dard tableau of shape λ with respect to the totally order in Definition2.1.Moreover SSTab( λ [ µ ]) denotes the set of semi standard tableaux ofshape λ [ µ ] . Definition 3.2.
For given T ∈ SSTab( λ [ µ ]) we define the weight wt( T ) as usual, i.e. wt( T ) = ( ν , ν , · · · ) , where ν k is the number of entriesequal to k . For λ ⊢ m, µ ⊢ n and ν ⊢ mn , we put Y νλ [ µ ] := λ [ µ ]; ν ) . Example 3.3.
Set λ = (2) , µ = (2) and ν = (2 , , . Then the Youngtableaux of shape (2)[(2)] with weight (2 , , are as follows; Hence we have Y (2 , )(2)[(2)] = 2 . Example 3.4.
Set λ = (2 , , µ = (1 ) and ν = (3 , ) . Then theYoung tableaux of shape (2 , )] with weight (3 , ) are as follows; K. IIJIMA
11 1 11 12 34 23 4
Hence we have Y (3 , )(2 , )] = 2 . Now we prove the first main result in this paper.
Theorem 3.5.
Let λ ⊢ m, µ ⊢ n and ν ⊢ mn . Then Y νλ [ µ ] is equal tothe coefficient of m ν in the expansion of s λ [ s µ ] in terms of monomialsymmetric functions.In other words, we have s λ [ s µ ] = X ν ⊢ mn Y νλ [ µ ] m ν . Proof.
Before the proof, we introduce some notations. For a positiveinteger s , we set r = µ ) ≤ s . For 1 ≤ i ≤ r , let T i be the i -th largest semi standard tableau in SSTab( µ ) ≤ s and set y i = x T i . Inparticular, SSTab( µ ) ≤ s = { T , · · · , T r } . Note that there is a naturalbijection ι : SSTab( λ ) ≤ r −→ SSTab( λ, SSTab( µ ) ≤ s ) −→ SSTab( λ [ µ ]) ≤ s such that y U = x ι ( U ) for U ∈ SSTab( λ ) ≤ r . For example, if λ = (1 ), µ = (2), T = 1 1 , T = 1 2 and U = 12 , then ι ( U ) =1 21 1 , y = x T = x , y = x T = x x , y U = y y and x ι ( U ) = x x .By Corollary 2.7, in the case of s -variables we have; s µ ( x , · · · , x s ) = X T ∈ SSTab( µ ) ≤ s x T = y + y + · · · + y r . HE FIRST TERM OF PLETHYSMS 7
Thus by Proposition 2.8, we have; s λ [ s µ ]( x , · · · , x s ) = s λ ( x T , · · · , x T r ) = s λ ( y , · · · , y r )= X U ∈ SSTab( λ ) ≤ r y U = X ι ( U ) ∈ SSTab( λ [ µ ]) ≤ s x ι ( U ) . Here by taking the limit s → ∞ , we have the following equality assymmetric function; s λ [ s µ ] = X T ∈ SSTab( λ [ µ ]) x T = X ν ⊢ mn Y νλ [ µ ] m ν . (cid:3) The first term of plethysm
Definition 4.1.
Let λ ⊢ m and µ ⊢ n . The first term of the plethysm s λ [ s µ ] is the maximal element in the set { ν ⊢ mn | a νλ [ µ ] = 0 } with respectto the reverse lexicographic order. Theorem 4.2. ([2;Conjecture 2],[3;Conjecture 1.2] and [12;Conjecture5.1])
Let λ ⊢ m , µ ⊢ n , l = l ( λ ) and l ′ = l ( µ ) . ( Where l = l ( λ ) is the lengthof partition λ . ) Then the first term of plethysm s λ [ s µ ] is ν := ( mµ , mµ , · · · , m ( µ l ′ −
1) + λ , λ , · · · , λ l ) . Moreover the coefficient of the first term is equal to . l − l ′ λ ′ move λ ′ = ( λ , · · · , λ l ) mµ mµ l ′ Proof.
By proposition 2.3, note that s ν = X κ K − κ,ν h κ . K. IIJIMA
Then we have; a νλ [ µ ] = h s λ [ s µ ] , s ν i = h s λ [ s µ ] , X κ K − κ,ν h κ i = X κ K − κ,ν h s λ [ s µ ] , h κ i = X κ K − κ,ν Y κλ [ µ ] , ( by Theorem 3.5 and property of h , i )= Y νλ,µ + X κ>ν K − κ,ν Y κλ [ µ ] , (by Corollary 2.6 (ii))Thus the assertion follows from the next lemma. Lemma 4.3. (1) max { ν ⊢ mn | Y νλ [ µ ] = 0 } = ν . (2) Y ν λ [ µ ] = 1 .Proof. We define a total order on the set of monomials in x i ’s ( i =1 , , · · · ) by lexicographic order. (For example, x < x x < x x Thus the sequence T (1) ≥ T (2) ≥ · · · ≥ T ( m ) that have a maximalweight under the condition Y νλ [ µ ] = 0 is only T (1) = · · · = T ( λ ) = T T ( λ +1) = · · · = T ( λ + λ ) = T ... ... ... T ( m − λ l +1) = · · · = T ( m ) = T l . Therefore the maximal weight iswt( T (1) ) + · · · + wt( T ( m ) ) = λ wt( T ) + · · · + λ l wt( T l )= ( mµ , mµ , m ( µ l ′ − 1) + λ , λ , · · · , λ l )= ν (cid:3)(cid:3) In particular, from the first half of this proof we get a new com-binatorial formula for plethysm. (Note that we can also express K − κ,ν combinatorially.[6] ) Corollary 4.4. a νλ [ µ ] = X κ ⊢ mn K − κ,ν Y κλ [ µ ] . Example 4.5. We calculate a (2 , from this formula. Since K − , , (2 , =1 , K − , , (2 , = − , K − , (2 , = 0 , Y (2 , = 2 and Y (3 , = 1 , we have a (2 , = K − , , (2 , Y (2 , + K − , , (2 , Y (3 , + K − , (2 , Y (4)(2)[(2)] = 2 − . Some remarks Our purpose for plethysm is stated as follows. Problem . Express the expansion coefficients a νλ [ µ ] combinatorially.For this problem, by imitating a proof of Littlewood-Richardson rulegiven in [7], we have the followings.Let l = l ( ν ). By Jacobi-Trudi’s formula, we have s ν = X π ∈ S l sgn( π ) h π ∗ ν , ( π ∗ ν = ( ν π ( i ) − π i + i ) ≤ i ≤ l ) . Here by Proposition 3.5, we have a νλ [ µ ] = h s λ [ s µ ] , s ν i = X π ∈ S l sgn( π ) h s λ [ s µ ] , h π ∗ ν i = X π ∈ S l sgn( π ) Y π ∗ νλ [ µ ] . So set A = { ( π, T ) | π ∈ S l , T ∈ SSTab( λ [ µ ]; π ∗ ν ) } , then we have a νλ [ µ ] = X ( π,T ) ∈ A sgn( π ) . Therefore the following conjecture is expected. Conjecture 5.1. There are a subset S ⊂ SSTab( λ [ µ ]) and a bijectivemap φ : A − A ∋ ( π, T ) → ( π ′ , T ′ ) ∈ A − A such that sgn( π ) = − sgn( π ′ ) , where A = { ( π, T ) ∈ A | T ∈ S } . In the case of Littlewood-Richardson rule, it is possible to take ”theset of lattice permutations” as A . Then φ can be defined ” properly ”.([7])Indeed, the following property holds. Lemma 5.2. For any λ ⊢ m, µ ⊢ n and ν ⊢ mn , we have Y νλ [ µ ] ≥ a νλ [ µ ] . Proof. Recall s λ [ s µ ] = X κ ⊢ mn a κλ [ µ ] s κ , and comparing the coefficients of the monomial symmetric function m ν ,we have Y νλ [ µ ] = X κ ⊢ mn a κλ [ µ ] K κ,ν = a νλ [ µ ] + X κ>ν a κλ [ µ ] K κ,ν , ( by Corollary 2.6 (i) and K ν,ν = 1) ≥ a νλ [ µ ] , ( a κλ [ µ ] ≥ K κ,ν ≥ . (cid:3) HE FIRST TERM OF PLETHYSMS 11 References [1] Y.Agaoka, On the curvature of Riemannian submanifolds of codimension 2 ,Hokkaido Math. J.14(1985),107-135.[2] Y.Agaoka, an algorithm to calculate plethysms of Schur functions and the tableup to total degree 16 , Technical Report No.46,Hiroshima University,1995.[3] Yoshio Agaoka, Combinatorial Conjectures on the Range of Young diagramsappearing in Plethysms , Technical Report No.59,Hiroshima University,1998.[4] Carr´ e and Leclerc, Splitting the square of a Schur function into its symmetricparts and antisymmetric parts , J.Algebraic Combinatorics, (1995), 201-231.[5] Y.M.Chen,A.M.Garsia and J.B.Remmel, Algorithms for Plethysm ,Contemp.Math. (1984)109-153.[6] O.Egecioglu and J.B.Remmel, A Combinatorial Interpretation of the InverseKostka Matrix , Lin. and Multi.Lin.Alg. (1990)59-84.[7] V.Gasharov, A short proof of the Littlewood-Richardson rule , EurepeanJ.Combin., 19(1998),451-453.[8] G.James and A.Kerber, The Representation Theory of the SymmetricGroup , Encyclopedia of Mathematics and its Applications, Vol. , Addison-Wesley,Reading,Mass.,1981.[9] D.E.Littlewood, Polynomial Concomitants And Invariant Matrices , J. LondonMath.Soc. (1936)49-55.[10] I.G.Macdonald, Symmetric Functions and Hall Polynomials second edition ,Oxford Mathematical Monographs,1995.[11] M.Yang, An Algorithm for Computing Plethysm Coefficients , Disc.Math. (1998)391-402.[12] M.Yang, The First Term in the Expansion of Plethysm of Schur Functions ,Disk.Math. (2002)331-341. Graduate School of Mathematics Nagoya University, Chikusa-kuNagoya 464-8602 Japan E-mail address ::