Schur type functions associated with polynomial sequences of binomial type
aa r X i v : . [ m a t h . R T ] N ov SCHUR TYPE FUNCTIONS ASSOCIATED WITHPOLYNOMIAL SEQUENCES OF BINOMIAL TYPE
MINORU ITOH
Abstract.
We introduce a class of Schur type functions associated with polynomialsequences of binomial type. This can be regarded as a generalization of the ordinarySchur functions and the factorial Schur functions. This generalization satisfies someinteresting expansion formulas, in which there is a curious duality. Moreover this classincludes examples which are useful to describe the eigenvalues of Capelli type centralelements of the universal enveloping algebras of classical Lie algebras.
Introduction
In this article, we introduce a class of Schur type functions associated with polynomialsequences of binomial type. Namely, inspired by the definition of the ordinary Schurfunction det( x λ i + N − ij ) / det( x N − ij ), we consider the following Schur type function:det( p λ i + N − i ( x j )) / det( p N − i ( x j )) . Here { p n ( x ) } n ≥ is a polynomial sequence of binomial type. This can be regarded as ageneralization of the ordinary Schur functions and of the factorial Schur functions ([BL],[CL]). We also consider the following function:det( p ∗ λ i + N − i ( x j )) / det( p ∗ N − i ( x j )) . Here we put p ∗ n ( x ) = x − p n +1 ( x ) (this is a polynomial, and satisfies good relations; seeSection 1.3). The main results of this article are some expansion formulas for thesefunctions and their mysterious duality corresponding to the exchange p n ( x ) ↔ p ∗ n ( x ) andthe conjugation of partitions (Sections 3–6). Most of them are proved by elementaryand straightforward calculations. Besides these results, we also give an application torepresentation theory of Lie algebras (Section 8).Let us briefly explain this application. The factorial Schur functions are useful to ex-press the eigenvalues of Capelli type central elements of the universal enveloping algebrasof the general linear Lie algebra (more precisely, we should say that the “shifted Schurfunctions” are useful; by the shift of variables, the factorial Schur functions are trans-formed into the shifted Schur functions ([OO1], [O])). In this article, we aim to introducesimilar Schur type functions which is useful to express the eigenvalues of Capelli typecentral elements of the universal enveloping algebras of the orthogonal and symplecticLie algebras. This aim is achieved in the case of the polynomial sequence correspondingto the central difference. This case associated with the central difference is also relatedwith the analogues of the shifted Schur functions given in [OO2], which were introduced Mathematics Subject Classification.
Primary 05E05, 05A40; Secondary 17B35, 15A15;
Key words and phrases.
Schur functions, polynomial sequence of binomial type, central elements ofuniversal enveloping algebras.This research was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 17740080. with a similar aim. Moreover, from this investigation of eigenvalues, we see the relationbetween following two central elements of U ( o N ) and U ( sp N ) (Theorem 8.6): (a) the cen-tral elements in terms of the column-determinant and the column-permanent given in [W]and [I6], (b) analogues of the quantum immanants given in [OO2]. Our class is a naturalgeneralization of the Schur functions containing these interesting functions related withthe classical Lie algebras.Various generalizations are known for the Schur functions. Many of them are obtainedby replacing the ordinary powers by some polynomial sequence (further generalizationsare known; see [M2]). In particular, the generalization associated with the polynomials inthe form p n ( x ) = Q nk =1 ( x − a k ) is well known [M1], and this contains the factorial Schurfunction. In this article, we consider another generalization which is not particularly largebut includes interesting phenomena and examples.1. Polynomial sequences of binomial type
First, we recall the properties of polynomial sequences of binomial type. See [MR], [R],[RKO], and [S] for further details.1.1. We start with the definition. A polynomial sequence { p n ( x ) } n ≥ in which the degreeof each polynomial is equal to its index is said to be of binomial type when the followingrelation holds for any n ≥ p n ( x + y ) = X k ≥ (cid:18) nk (cid:19) p k ( x ) p n − k ( y ) . Let us see some examples. First, the sequence { x n } n ≥ of the ordinary powers is ofbinomial type, because we have the relation( x + y ) n = X k ≥ (cid:18) nk (cid:19) x k y n − k (the ordinary binomial expansion). As other typical examples, some factorial powers arewell known. We define the rising factorial power x n and falling factorial power x n by x n = x ( x + 1) · · · ( x + n − , x n = x ( x − · · · ( x − n + 1) . Then { x n } n ≥ and { x n } n ≥ are also of binomial type. Indeed the following relations hold([MR], [RKO]):( x + y ) n = X k ≥ (cid:18) nk (cid:19) x k y n − k , ( x + y ) n = X k ≥ (cid:18) nk (cid:19) x k y n − k . It is easily seen that p n (0) = δ n, , when { p n ( x ) } n ≥ is of binomial type.1.2. A natural correspondence is known between polynomial sequences of binomial typeand delta operators [RKO].We recall the definition of delta operators. A linear operator Q = Q x : C [ x ] → C [ x ] iscalled a delta operator when the following two properties hold: (i) Q reduces degrees ofpolynomials by one; (ii) Q is shift-invariant (that is, Q commutes with all shift operators E a : f ( x ) f ( x + a )). A typical example is the differentiation D = ddx . Moreover the forward difference ∆ + and the backward difference ∆ − are delta operators:∆ + : f ( x ) f ( x + 1) − f ( x ) , ∆ − : f ( x ) f ( x ) − f ( x − . CHUR FUNCTIONS FOR POLYNOMIAL SEQUENCES OF BINOMIAL TYPE 3
Every delta operator can be written as a power series in the differentiation operator D ofthe following form with a , a , . . . ∈ C , a = 0: Q = a D + a D + a D + · · · . There is a natural one-to-one correspondence between these delta operators and poly-nomial sequences of binomial type. These are related via the relation(1.1) Qp n ( x ) = np n − ( x ) . Namely, for a polynomial sequence of binomial type { p n ( x ) } n ≥ , the linear operator Q : C [ x ] → C [ x ] determined by (1.1) is a delta operator. Conversely, for any delta opera-tor Q , a polynomial sequence { p n ( x ) } n ≥ is uniquely determined by (1.1) and the relation p n (0) = δ n, , and this sequence is of binomial type (these are called basic polynomials ).For example, the differentiation D = ddx corresponds to the sequence { x n } n ≥ , because Dx n = nx n − . Similarly, the forward difference ∆ + and the backward difference ∆ − correspond to the sequences { x n } n ≥ and { x n } n ≥ , respectively:∆ − x n = nx n − , ∆ + x n = nx n − . { p n ( x ) } n ≥ of binomial type, weput p ∗ n ( x ) = x − p n +1 ( x ) . This is a polynomial, because the constant term of p n +1 ( x ) is equal to 0 if n ≥ p ∗ n ( x ) satisfies the following relations (we can prove this by induction). Inother words, { p ∗ n ( x ) } n ≥ is a Sheffer sequence [R]. Proposition 1.1.
We have Qp ∗ n ( x ) = np ∗ n − ( x ) , p ∗ n ( x + y ) = X k ≥ (cid:18) nk (cid:19) p k ( x ) p ∗ n − k ( y ) = X k ≥ (cid:18) nk (cid:19) p ∗ k ( x ) p n − k ( y ) . These polynomials can be extended naturally for n < C (( x − )) = { P k ≤ n a k x k | a k ∈ C , n ∈ Z } . Namely we have the following proposition (this is alsoproved by induction): Proposition 1.2.
Let Q be a delta operator. Then there uniquely exist { p n ( x ) } n ∈ Z and { p ∗ n ( x ) } n ∈ Z satisfying the relations Qp n ( x ) = np n − ( x ) , Qp ∗ n ( x ) = np ∗ n − ( x ) , p n +1 ( x ) = xp ∗ n ( x ) and p n ( x + y ) = X k ≥ (cid:18) nk (cid:19) p k ( x ) p n − k ( y ) ,p ∗ n ( x + y ) = X k ≥ (cid:18) nk (cid:19) p k ( x ) p ∗ n − k ( y ) = X k ≥ (cid:18) nk (cid:19) p ∗ k ( x ) p n − k ( y ) . Here we regard the last two relations as equalities in C [ x ](( y − )) . MINORU ITOH
Note that p ∗− ( x ) must be equal to x − , because xp ∗− ( x ) = p ( x ) = 1. Thus, thisextension is unique.From now on, we denote these polynomials associated with the delta operator Q by p n ( x ) = p Qn ( x ) and p ∗ n ( x ) = p ∗ Qn ( x ).The polynomial p ∗ n ( x ) is not a mere supplementary object, but plays a role as importantas p n . We will exhibit some dualities between these two polynomials.1.4. Consider the following operator: R x = [ Q x , x ] = Q x x − xQ x . We can easily see that R x p ∗ k ( x ) = p k ( x ), and R x is invertible and shift-invariant. Let usput p ( a ) k ( x ) = R ax p k ( x ) for a ∈ Z , so that p ( − k ( x ) = p ∗ k ( x ). As seen by induction, this p ( a ) k ( x ) satisfies the relation p ( a + b ) n ( x + y ) = X k ≥ (cid:18) nk (cid:19) p ( a ) n − k ( x ) p ( b ) k ( y ) . In the case of Q = ∆ + , this operator R x maps f ( x ) to f ( x + 1) (that is, p ( a ) n ( x ) =( x + a − n ), and this is an algebra automorphism on C [ x ], but this is not true for general Q . See also Remark in Section 2.1.5. In the remainder of this article, we assume that Q is normalized in the sense thatthe coefficient of D is equal to 1: Q = D + a D + a D + · · · . Under this assumption, the associated polynomials p n ( x ) = p Qn ( x ) and p ∗ n ( x ) = p ∗ Qn ( x )become monic. Conversely, the delta operator associated with a monic polynomial se-quence of binomial type is automatically normalized. Thus the following assumptions areequivalent: (i) Q is normalized; (ii) p Qn ( x ) is monic; (iii) p ∗ Qn ( x ) is monic. This assumptionis not an essential one, but merely for simplicity.1.6. Let us see some fundamental examples.(1) In the case Q = D = ddx , we have p n ( x ) = x n and p ∗ n ( x ) = x n .(2) In the case Q = ∆ + , we have p n ( x ) = x n , and hence p ∗ n ( x ) = ( x − n .(3) In the case Q = ∆ − , we have p n ( x ) = x n , and hence p ∗ n ( x ) = ( x + 1) n .(4) We define the central difference ∆ by∆ f ( x ) = f ( x + ) − f ( x − ) . This is also a delta operator. In the case Q = ∆ , we have p ∗ n ( x ) = x n . Here we put x n = ( x + n − )( x + n − ) · · · ( x − n − ) . Hence, p n ( x ) is expressed as p n ( x ) = x · x n . This is seen by a direct calculation.See [R], [RKO], and [S] for other examples (Abel polynomials, Laguerre polynomials,etc.). CHUR FUNCTIONS FOR POLYNOMIAL SEQUENCES OF BINOMIAL TYPE 5 Definition of Schur type functions
Let us define our main objects, the Schur type functions associated with a polyno-mial sequence of binomial type. Let p n ( x ) = p Qn ( x ) and p ∗ n ( x ) = p ∗ Qn ( x ) be polynomialscorresponding to a normalized delta operator Q .For λ = ( λ , . . . , λ N ) ∈ Z N , we consider the following determinants:˜ s Qλ ( x , . . . , x N ) = det p λ + N − ( x ) . . . p λ + N − ( x N ) p λ + N − ( x ) . . . p λ + N − ( x N )... ... p λ N +0 ( x ) . . . p λ N +0 ( x N ) , ˜ s ∗ Qλ ( x , . . . , x N ) = det p ∗ λ + N − ( x ) . . . p ∗ λ + N − ( x N ) p ∗ λ + N − ( x ) . . . p ∗ λ + N − ( x N )... ... p ∗ λ N +0 ( x ) . . . p ∗ λ N +0 ( x N ) . We regard these as elements of C (( x − , . . . , x − N )) = ( X k ≤ n ,...,k N ≤ n N a k ,...,k N x k · · · x k N N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a k ,...,k N ∈ C , n , . . . , n N ∈ Z ) . It is easy to see that these two determinants are alternating in x , . . . , x N . Moreover,when λ = ∅ = (0 , . . . , s Q ∅ ( x , . . . , x N ) = ˜ s ∗ Q ∅ ( x , . . . , x N ) = ∆( x , . . . , x N ) = Y ≤ i 0. When λ = ( λ , . . . , λ N )is a partition, namely when λ ≥ · · · ≥ λ N ≥ 0, we can regard λ as a Young diagram.In this case, the polynomials s λ (respectively, s ∗ λ ) form a basis of the space of symmetricpolynomials.We can also consider the counterparts of elementary symmetric functions and completehomogeneous symmetric functions (note that h k and h ∗ k can be defined even if k is a MINORU ITOH negative integer): e k ( x , . . . , x N ) = s (1 k ) ( x , . . . , x N ) ,e ∗ k ( x , . . . , x N ) = s ∗ (1 k ) ( x , . . . , x N ) ,h k ( x , . . . , x N ) = s ( k ) ( x , . . . , x N ) ,h ∗ k ( x , . . . , x N ) = s ∗ ( k ) ( x , . . . , x N ) . Here we used the abbreviation( a m , . . . , a m n n ) = ( m times z }| { a , . . . , a , . . . , m n times z }| { a n , . . . , a n , , . . . , . These functions are not equal to the ordinary elementary symmetric functions and theordinary complete symmetric functions in general, but there are two exceptions. Namely e N and h ∗− N are independent of { p n ( x ) } . Proposition 2.1. We have e N ( x , . . . , x N ) = x · · · x N , h ∗− N ( x , . . . , x N ) = 1 x · · · x N . This is easy from the following more general relation: Proposition 2.2. We have s ( λ ,...,λ N ) ( x , . . . , x N ) = s ∗ ( λ − ,...,λ N − ( x , . . . , x N ) · x · · · x N . This is immediate from˜ s ( λ ,...,λ N ) ( x , . . . , x N ) = ˜ s ∗ ( λ − ,...,λ N − ( x , . . . , x N ) · x · · · x N . The following relation between s and s ∗ is also confirmed by a direct calculation: Proposition 2.3. When λ i ≥ , we have s ∗ ( λ ,...,λ N ) ( x , . . . , x N ) = s ( λ ,...,λ N ) ( x , . . . , x N , . Remark. The shifted Schur function is defined as follows [OO1]:det(( x j + N − j ) λ i + N − i ) / det(( x j + N − j ) N − i ) . In the case of Q x = ∆ + x , we have p n ( x ) = x n , and R x = [ Q x , x ] = Q x x − xQ x is equal tothe algebra automorphism f ( x ) f ( x + 1). Thus this function can be expressed asdet( p ( N − j ) λ i + N − i ( x j )) / det( p ( N − j ) N − i ( x j ))= R N − x R N − x · · · R x N det( p λ i + N − i ( x j )) / det( p N − i ( x j )) . Noting this and Proposition 2.3, we can naturally consider the projective limit of thisfunction. This is an advantage of considering the shift x j x j + N − j .How about the case of an arbitrary delta operator Q x ? In general, we do not havesuch a good relation, because R x is not an algebra automorphism (thus this is not apolynomial in general, even if λ is a partition). Thus it does not seem easy to consider anatural infinite-variable version for general Q x . CHUR FUNCTIONS FOR POLYNOMIAL SEQUENCES OF BINOMIAL TYPE 7 Expansions of Schur type functions For the Schur type functions defined in the previous section, we have the followingexpansions (this can be regarded as a generalization of Example 10 in Section I.3 in[M1]): Theorem 3.1. For λ ≥ λ ≥ · · · ≥ λ N , we have s λ ( x + u, . . . , x N + u ) = X µ ⊂ λ d λµ ( u ) s µ ( x , . . . , x N ) ,s ∗ λ ( x + u, . . . , x N + u ) = X µ ⊂ λ d λµ ( u ) s ∗ µ ( x , . . . , x N ) ,s ∗ λ ( x + u, . . . , x N + u ) = X µ ⊂ λ d ∗ λµ ( u ) s µ ( x , . . . , x N ) as equalities in C [ u ](( x − , . . . , x − N )) . Here µ runs over µ = ( µ , . . . , µ N ) such that µ ≥ µ ≥ · · · ≥ µ N , µ ≤ λ , . . . , µ N ≤ λ N , and d λµ ( u ) and d ∗ λµ ( u ) are defined by d λµ ( u ) = det (cid:18)(cid:18) λ i + N − iλ i − µ j − i + j (cid:19) p λ i − µ j − i + j ( u ) (cid:19) ≤ i,j ≤ N ,d ∗ λµ ( u ) = det (cid:18)(cid:18) λ i + N − iλ i − µ j − i + j (cid:19) p ∗ λ i − µ j − i + j ( u ) (cid:19) ≤ i,j ≤ N . To prove this, we use the Cauchy–Binet formula: Proposition 3.2. We have det( AB ) ( i ,...,i k ) , ( j ,...,j k ) = X ≤ r < ··· Thus A is expressed as A = BC with the N × ∞ matrix B and the ∞ × N matrix C defined by B = (cid:0) l (cid:1) p ( u ) (cid:0) l (cid:1) p ( u ) . . . (cid:0) l l − l (cid:1) p l − l ( u ) (cid:0) l l − l +1 (cid:1) p l − l +1 ( u ) . . . ... ... (cid:0) l N l N − l (cid:1) p l N − l ( u ) (cid:0) l N l N − l +1 (cid:1) p l N − l +1 ( u ) . . . ,C = p l ( x ) p l ( x ) . . . p l ( x N ) p l − ( x ) p l − ( x ) . . . p l − ( x N )... ... ... . Applying the Cauchy–Binet formula (Proposition 3.2) to this, we have˜ s λ ( x + u, . . . , x N + u )= det A = X k ,...,k N det (cid:0) l l − k (cid:1) p l − k ( u ) (cid:0) l l − k (cid:1) p l − k ( u ) . . . (cid:0) l l − k N (cid:1) p l − k N ( u ) (cid:0) l l − k (cid:1) p l − k ( u ) (cid:0) l l − k (cid:1) p l − k ( u ) . . . (cid:0) l l − k N (cid:1) p l − k N ( u )... ... ... (cid:0) l N l N − k (cid:1) p l N − k ( u ) (cid:0) l N l N − k (cid:1) p l N − k ( u ) . . . (cid:0) l N l N − k N (cid:1) p l N − k N ( u ) · det p k ( x ) p k ( x ) . . . p k ( x N ) p k ( x ) p k ( x ) . . . p k ( x N )... ... ... p k N ( x ) p k N ( x ) . . . p k N ( x N ) = X µ ⊂ λ d λµ ( u )˜ s µ ( x , . . . , x N ) . Here, the first sum is over k , . . . , k N satisfying k i ≤ l i and k > · · · > k N , and the secondover µ , . . . , µ N satisfying µ i ≤ λ i and µ ≥ · · · ≥ µ N (we define µ i by k i = µ i + N − i ).Dividing this by ∆( x + u, . . . , x N + u ) = ∆( x , . . . , x N ), we have the assertion. (cid:3) It is interesting to consider the following variant of d λµ ( u ) when λ and µ are partitions:ˆ d λµ ( u ) = Q j ( µ j + N − j )! Q i ( λ i + N − i )! d λµ ( u ) = det (cid:0) p ( λ i − µ j − i + j ) ( u ) (cid:1) ≤ i,j ≤ N . Here we put p ( n ) ( x ) = n ! p n ( x ) suggested by the notation of divided power x ( n ) = n ! x n . Itis easily seen that ˆ d λµ ( u ) is independent of N , though d λµ ( u ) depends on N . Namely, ˆ d λµ does not change, even if we append some zeros at the ends of λ and µ . For this ˆ d λµ ( u ),the following duality holds: Theorem 3.3. For two partitions λ and µ , we have ˆ d λµ ( u ) = ( − ) | λ |−| µ | ˆ d λ ′ µ ′ ( − u ) . Here λ ′ and µ ′ mean the conjugates of λ and µ , respectively. This theorem follows from Theorem 6.1 below. It can also be deduced from the fol-lowing relation (essentially the same as (2.9) in [M1]), because P k ≥ p ( k ) ( u ) p ( n − k ) ( − u ) = p ( n ) (0) = δ n, : CHUR FUNCTIONS FOR POLYNOMIAL SEQUENCES OF BINOMIAL TYPE 9 Theorem 3.4. Assume that sequences { c k } k ≥ and { c ′ k } k ≥ satisfy the relations c = c ′ = 1 and P k ≥ ( − ) k c k c ′ n − k = δ n, . Then, for two partitions λ and µ , we have det (cid:0) c λ i − µ j − i + j (cid:1) ≤ i,j ≤ depth λ = det (cid:16) c ′ λ ′ i − µ ′ j − i + j (cid:17) ≤ i,j ≤ depth λ ′ . Here we interpret c n and c ′ n as for n < . Expansions of e and h The expansion formulas for the functions e k , e ∗ k , h k , and h ∗ k have more interestingaspects. Some of these formulas are deduced from the results in the previous sectionas special cases, but the others are not, and we observe a mysterious duality in theseformulas. For simplicity, we introduce the following notation: h k ( x , . . . , x N ; u ) = h k ( x + u, . . . , x N + u ) ,h ∗ k ( x , . . . , x N ; u ) = h ∗ k ( x + u, . . . , x N + u ) ,e k ( x , . . . , x N ; u ) = e k ( x − u, . . . , x N − u ) ,e ∗ k ( x , . . . , x N ; u ) = e ∗ k ( x − u, . . . , x N − u ) . These can be expanded as follows: Theorem 4.1. For k ≥ , we have e k ( x , . . . , x N ; u ) = X l ≥ (cid:18) − N + k − k − l (cid:19) e l ( x , . . . , x N ) p k − l ( u )= X l ≥ (cid:18) − N + k − k − l (cid:19) e ∗ l ( x , . . . , x N ) p ∗ k − l ( u ) ,e ∗ k ( x , . . . , x N ; u ) = X l ≥ (cid:18) − N + k − k − l (cid:19) e ∗ l ( x , . . . , x N ) p k − l ( u ) . Theorem 4.2. For k ≥ , we have h k ( x , . . . , x N ; u ) = X l ≥ (cid:18) N + k − k − l (cid:19) h l ( x , . . . , x N ) p k − l ( u ) ,h ∗ k ( x , . . . , x N ; u ) = X l ≥ (cid:18) N + k − k − l (cid:19) h ∗ l ( x , . . . , x N ) p k − l ( u )= X l ≥ (cid:18) N + k − k − l (cid:19) h l ( x , . . . , x N ) p ∗ k − l ( u ) . Comparing these two theorems, we observe a duality corresponding to the exchanges e ↔ h ∗ and e ∗ ↔ h . Note that the following are not equalities in general: e ∗ k ( x , . . . , x N ; u ) = X l ≥ (cid:18) − N + k − k − l (cid:19) e l ( x , . . . , x N ) p ∗ k − l ( u ) ,h k ( x , . . . , x N ; u ) = X l ≥ (cid:18) N + k − k − l (cid:19) h ∗ l ( x , . . . , x N ) p ∗ k − l ( u ) . Theorem 4.2 can be extended for negative integers k as follows: Theorem 4.3. For k ∈ Z , we have h k ( x , . . . , x N ; u ) = X l ≥ (cid:18) N + k − N + l − (cid:19) h l ( x , . . . , x N ) p k − l ( u ) ,h ∗ k ( x , . . . , x N ; u ) = X l ≥ (cid:18) N + k − N + l − (cid:19) h ∗ l ( x , . . . , x N ) p k − l ( u )= X l ≥ (cid:18) N + k − N + l − (cid:19) h l ( x , . . . , x N ) p ∗ k − l ( u ) . Theorem 4.4. For k ∈ Z , we have h k ( x , . . . , x N ; u ) = X l ≥ (cid:18) N + k − l (cid:19) h k − l ( x , . . . , x N ) p l ( u ) ,h ∗ k ( x , . . . , x N ; u ) = X l ≥ (cid:18) N + k − l (cid:19) h ∗ k − l ( x , . . . , x N ) p l ( u )= X l ≥ (cid:18) N + k − l (cid:19) h k − l ( x , . . . , x N ) p ∗ l ( u ) . We can prove Theorems 4.2–4.4 easily in a way similar to the general expansion Theo-rem 3.1. The proof of Theorem 4.1 is as follows: Proof of Theorem . First we prove the case of k = N . Noting (2.1), we have˜ s (1 N ) ( x − u, . . . , x N − u )= det p N ( x − u ) . . . p N ( x N − u )... ... p ( x − u ) . . . p ( x N − u ) = ( x − u ) · · · ( x N − u ) det p ∗ N − ( x − u ) . . . p ∗ N − ( x N − u )... ... p ∗ ( x − u ) . . . p ∗ ( x N − u ) = ( x − u ) · · · ( x N − u )∆( x − u, . . . , x N − u )= ( x − u ) · · · ( x N − u )∆( x , . . . , x N )= ∆( x , . . . , x N , u )= det p N ( x ) . . . p N ( x N ) p N ( u )... ... ... p ( x ) . . . p ( x N ) p ( u ) . By the cofactor expansion along the last column, we see that this is equal to N X k =0 ( − ) k ˜ s (1 N − k ) ( x , . . . , x N ) p k ( u ) . CHUR FUNCTIONS FOR POLYNOMIAL SEQUENCES OF BINOMIAL TYPE 11 Dividing both sides by the difference product ∆( x − u, . . . , x N − u ) = ∆( x , . . . , x N ), wehave e N ( x − u, . . . , x N − u ) = X k ≥ ( − ) k p k ( u ) e N − k ( x , . . . , x N ) . Thus we have the assertion for k = N .The other cases are deduced from this. Indeed, on one hand, we have e N ( x − u − w, . . . , x N − u − w ) = X l ≥ ( − ) l p l ( w ) e N − l ( x − u, . . . , x N − u ) , and on the other hand e N ( x − u − w, . . . , x N − u − w ) = X k ≥ ( − ) k p k ( u + w ) e N − k ( x , . . . , x N )= X k ≥ X l ≥ ( − ) k (cid:18) kl (cid:19) p l ( w ) p k − l ( u ) e N − k ( x , . . . , x N ) . Comparing the coefficients of p l ( w ), we have the general case. (cid:3) The following relation for the delta operator is easy to deduce from these expansions: Corollary 4.5. We have Q u h k ( x , . . . , x N ; u ) = ( N + k − h k − ( x , . . . , x N ; u ) ,Q u h ∗ k ( x , . . . , x N ; u ) = ( N + k − h ∗ k − ( x , . . . , x N ; u ) ,Q u e k ( x , . . . , x N ; u ) = ( − N + k − e k − ( x , . . . , x N ; u ) ,Q u e ∗ k ( x , . . . , x N ; u ) = ( − N + k − e ∗ k − ( x , . . . , x N ; u ) . Generating functions Combining Proposition 2.1 with the relations in the previous section, we have thefollowing relations (put k = N in Theorem 4.1 and k = − N in Theorems 4.3 and 4.4): Theorem 5.1. We have ( u − x ) · · · ( u − x N ) = X l ≥ ( − ) l e l ( x , . . . , x N ) p N − l ( u )= X l ≥ ( − ) l e ∗ l ( x , . . . , x N ) p ∗ N − l ( u ) . Theorem 5.2. We have u + x ) · · · ( u + x N ) = X l ≥ ( − ) l h l ( x , . . . , x N ) p ∗− N − l ( u )= X l ≥ ( − ) l h ∗ l ( x , . . . , x N ) p − N − l ( u ) . Theorem 5.3. We have u + x ) · · · ( u + x N ) = ( − ) N − X l ≥ ( − ) l h − N − l ( x , . . . , x N ) p ∗ l ( u )= ( − ) N − X l ≥ ( − ) l h ∗− N − l ( x , . . . , x N ) p l ( u ) . We can regard the left hand sides of these equalities as “generating functions” of e k , e ∗ k , h k , and h ∗ k represented as sums of multiples of p n ( x ) or p ∗ n ( x ) instead of the ordinarypower. Remark. The conclusion of Theorem 5.1 also holds, even if { p n ( u ) } is not of binomialtype. Namely, if p n ( u ) is a monic polynomial of degree n , we have( u − x ) · · · ( u − x N ) = X l ≥ ( − ) l e l ( x , . . . , x N ) p N − l ( u ) . This is easily seen from the proof of Theorem 4.1. Similarly, the assertions of Theorems 5.2and 5.3 also hold, if p n ( x ) and p ∗ n ( x ) are monic polynomials of degree n satisfying therelation 1 x + y = X k ≥ ( − ) k p k ( x ) p ∗− − k ( y ) = X k ≥ ( − ) k p ∗ k ( x ) p − − k ( y ) . Cauchy type relations The relations in the previous section can be generalized as analogues of the (dual)Cauchy identity. In this section, we often abbreviate a function f ( x , . . . , x N ) simplyas f ( x ). Theorem 6.1. We have Y ≤ i ≤ M Y ≤ j ≤ N ( y j − x i ) = X λ ( − ) | λ | s λ ( x ) s λ † ( y ) = X λ ( − ) | λ | s ∗ λ ( x ) s ∗ λ † ( y ) . Here λ runs over the Young diagrams satisfying depth( λ ) ≤ M and depth( λ ′ ) ≤ N .Moreover we define λ † by λ † = ( N − λ M , N − λ M − , . . . , N − λ ) ′ . Theorem 6.2. We have Y ≤ i ≤ M Y ≤ j ≤ N y j + x i = X λ ( − ) | λ | s ∗ λ ( x ) s λ ‡ ( y ) = X λ ( − ) | λ | s λ ( x ) s ∗ λ ‡ ( y ) in C [ x , . . . , x M ](( y − , . . . , y − N )) . Here λ runs over the Young diagrams λ satisfying depth( λ ) ≤ min( M, N ) . Moreover we define λ ‡ by λ ‡ = ( − M − λ N , − M − λ N − , . . . , − M − λ ) . From Theorem 6.1 we see the following duality:( − ) | λ | d λµ ( u ) = ( − ) | µ | d µ † λ † ( − u ) . Indeed, this follows by expanding Q ≤ i ≤ M Q ≤ j ≤ N y j + x i + u in two ways. Theorem 3.3 isimmediate from this. Remark. As in the previous section, the conclusion of Theorem 6.1 holds even if { p n ( u ) } isnot of binomial type. Namely it holds if p n ( u ) is a monic polynomial of degree n . Similarlythe conclusion of Theorem 6.2 also holds if p n ( x ) and p ∗ n ( x ) are monic polynomials ofdegree n satisfying1 x + y = X k ≥ ( − ) k p k ( x ) p ∗− − k ( y ) = X k ≥ ( − ) k p ∗ k ( x ) p − − k ( y ) . CHUR FUNCTIONS FOR POLYNOMIAL SEQUENCES OF BINOMIAL TYPE 13 Remark. We can regard Theorem 6.2 as a generalization of the following well-knownrelation (the Cauchy identity): Y ≤ i ≤ M, ≤ i ≤ N − x i y j = X λ s Dλ ( x ) s Dλ ( y ) . Proof of Theorem . By (2.1) we have∆( y )∆( x ) · Y ≤ i ≤ N Y ≤ j ≤ M ( y j − x i ) = ∆( y , . . . , y N , x , . . . , x M )= det p M + N − ( y ) . . . p M + N − ( y N ) p M + N − ( x ) . . . p M + N − ( x M ) p M + N − ( y ) . . . p M + N − ( y N ) p M + N − ( x ) . . . p M + N − ( x M )... ... ... ... p ( y ) . . . p ( y N ) p ( x ) . . . p ( x M ) . Applying the Laplace expansion to the first N columns, we see that this is equal to P λ ( − ) | λ | ˜ s λ ( x )˜ s λ † ( y ). Indeed we have { λ i + M − i | ≤ i ≤ M } ∪ { λ † j + N − j | ≤ j ≤ N } = { , , . . . , M + N − } (recall (1.7) in [M1]). This means the first equality. The second equality is similarlyshown by replacing p k by p ∗ k . (cid:3) To prove Theorem 6.2, we use the following well-known relation (the left hand side isknown as the Cauchy determinant ): Lemma 6.3. When M = N , we have det (cid:18) x i + y j (cid:19) = ∆( x )∆( y ) Q ≤ i,j ≤ N ( x i + y j ) . This relation is generalized as follows (this can be proved by induction): Lemma 6.4. When N ≥ M , we have det p ∗ N − M − ( y ) p ∗ N − M − ( y ) . . . p ∗ N − M − ( y N ) ... ... ... p ∗ ( y ) p ∗ ( y ) . . . p ∗ ( y N ) p ∗ ( y ) p ∗ ( y ) . . . p ∗ ( y N ) x + y x + y . . . x + y N x + y x + y . . . x + y N ... ... ... x M + y x M + y . . . x M + y N = ∆( x )∆( y ) Q ≤ i ≤ M, ≤ j ≤ N ( x i + y j ) . Proof of Theorem ( the case of M = N ). Using the Cauchy–Binet formula (Proposi-tion 3.2), we havedet (cid:18) x i + y j (cid:19) ≤ i,j ≤ N = det X k ≥ ( − ) k p k ( x i ) p ∗− − k ( y j ) ! ≤ i,j ≤ N = X ≤ k < ··· Note that det B (1 ,...,N ) , ( i ,...,i N ) = 0, unless ( i , . . . , i N − M ) = (1 , . . . , N − M ). Thus we havedet A = X ≤ k < ··· The proof in the case N < M is almost the same, so we omit it.7. Capelli type elements Our Schur type functions are useful to express the eigenvalues of Capelli type centralelements of the universal enveloping algebras of the classical Lie algebras. Before statingthis, we recall these Capelli type elements in this section.These central elements have been investigated in the study of Capelli type identities.See [HU], [MN], [O], [U1–5], [IU], and [I1–6] for the Capelli identity and its generalizations.7.1. First, we recall the Capelli elements, famous central elements of the universal en-veloping algebra U ( gl N ) of the general linear Lie algebra gl N . Let E ij be the standardbasis of gl N , and consider the matrix E = ( E ij ) ≤ i,j ≤ N in Mat N ( U ( gl N )). Then thefollowing determinant is known as the Capelli element ([Ca1], [HU], [U1]): C gl N ( u ) = det( E − u + diag ♮ N ) . Here is the unit matrix, and ♮ N is the sequence ♮ N = ( N − , N − , . . . , 0) of length N .Moreover “det” means a non-commutative determinant called the column-determinant. Namely, for a square matrix Z = ( Z ij ) whose entries are non-commutative, we putdet Z = X σ ∈ S N sgn( σ ) Z σ (1)1 Z σ (2)2 · · · Z σ ( N ) N . The Capelli element C gl N ( u ) is known to be central in U ( gl N ).This is generalized to the sums of minors C gl N k ( u ) = X ≤ i < ···
The elements C gl N k ( u ) and D gl N k ( u ) are also central in the universal enveloping algebra(actually these are generators of the center of the universal enveloping algebra; see [HU],[I4], [N], [U1] for the details). Theorem 7.1. For any u ∈ C , C gl N k ( u ) and D gl N k ( u ) are central in U ( gl N ) . These central elements act on the irreducible representations as scalar operators bySchur’s lemma. These values (the eigenvalues) can be calculated by noting the followingtriangular decomposition: gl N = n − ⊕ h ⊕ n + . Here n − , h , and n + are the subalgebras of gl N spanned by the elements F gl N ij such that i > j , i = j , and i < j , respectively. Namely, the entries in the lower triangular part, inthe diagonal part, and in the upper triangular part of the matrix E gl N belong to n − , h ,and n + , respectively.For example, we can calculate the eigenvalue of C gl N ( u ) as follows. Let π be the irre-ducible representation determined by the partition ( λ , . . . , λ N ), and consider the actionof C gl N ( u ) to the highest weight vector v : π ( C gl N ( u )) v = X σ ∈ S N sgn( σ ) π ( E σ (1)1 ( − u + N − · · · π ( E σ ( N ) N ( − u + 0)) v. Then, among these N ! terms, there remains only one term corresponding to σ = ( ... N ... N ),and the other N ! − π ( E ij ) v = 0 for i < j and π ( E ii ) v = λ i v .Thus we have π ( C gl N ( u )) v = π ( E ( − u + N − · · · π ( E NN ( − u + 0)) v = ( λ − u + N − · · · ( λ N − u + 0) v. and we see that C gl N ( u ) acts as multiplication by the scalar ( λ − u + N − · · · ( λ N − u +0)on π .We can write down the eigenvalue of C gl N k ( u ) similarly by considering the action to thehighest weight vector v of each column-determinant on the right hand side of (7.1). Indeed,also in this case, only one term remains of the k ! terms in each column-determinant.The calculation of the eigenvalue of D gl N k ( u ) is essentially same (but a bit more com-plicated). For this, we consider the action of each column-permanent on the right handside of (7.2) to v . Then, of the k ! terms in each column-permanent, there remain only I !terms corresponding to σ such that i σ ( a ) = i a for a = 1 , . . . , k , and these I ! terms are allequal to a scalar multiple of v .In this way, we can write down the eigenvalues of C gl N k ( u ) and D gl N k ( u ). However theresults are not so simple. As seen in Section 8 below, these eigenvalues are actuallyexpressed by using the factorial (shifted) Schur functions. Remark. We note some previous results:(1) We can also express the elements C gl N k ( u ) and D gl N k ( u ) in terms of the “symmetrizeddeterminant” and the “symmetrized permanents” ([IU], [I1–6]). From these expressions,we can easily see the centrality of these elements.(2) In [O], Okounkov introduced a class of central elements of U ( gl N ), and called them the quantum immanants. These elements S µ indexed by partitions µ are expressed in termsof a determinant type function called immanant, and form a basis of the center of U ( gl N ) as a vector. The central elements C gl N k (0) and D gl N k (0) can be regarded as the cases of µ = (1 k ) and µ = ( k ): C gl N k (0) = S (1 k ) , D gl N k (0) = S ( k ) . (7.3)Okounkov also gave a generalization of the Capelli identity (called higher Capelli identi-ties ) for the quantum immanants. In Section 8, we will see the eigenvalues of the quantumimmanants together with those of C gl N k ( u ) and D gl N k ( u ).7.2. Next, we recall analogues of the Capelli elements in the universal enveloping algebrasof the orthogonal Lie algebras given in [W].Let S ∈ Mat N ( C ) be a non-singular symmetric matrix of size N . We can realize theorthogonal Lie group as the isometry group with respect to the bilinear form determinedby S : O ( S ) = { g ∈ GL N | t gSg = S } . The corresponding Lie algebra is expressed as o ( S ) = { Z ∈ gl N | t ZS + SZ = 0 } . As generators of this o ( S ), we can take F o ( S ) ij = E ij − S − E ji S , where E ij is the standardbasis of gl N . We introduce the N × N matrix F o ( S ) whose ( i, j )th entry is this generator: F o ( S ) = ( F o ( S ) ij ) ≤ i,j ≤ N . We regard this matrix as an element of Mat N ( U ( o ( S ))).In particular, in the case of S = S = ( δ i,N +1 − j ), the corresponding orthogonal Liealgebra is expressed as follows: o ( S ) = { Z = ( Z ij ) ∈ gl N | Z ij + Z N +1 − j,N +1 − i = 0 } . We call this the split realization of the orthogonal Lie algebra. In this case, we can takethe following triangular decomposition: o ( S ) = n − ⊕ h ⊕ n + . Here n − , h , and n + are the subalgebras of o ( S ) spanned by the elements F o ( S ) ij such that i > j , i = j , and i < j , respectively. Namely, the entries in the lower triangular part, inthe diagonal part, and in the upper triangular part of the matrix E gl N belong to n − , h ,and n + , respectively. Thus, if there is a central element of U ( o ( S )) which is expressed asthe column-determinant of F o ( S ) , we can easily calculate its eigenvalue in a way similarto the case of C gl N ( u ). In fact, the following central element was given by Wachi [W]: Theorem 7.2 (Wachi) . For any u ∈ C , the following element is central in U ( o ( S )) : C o N ( u ) = det( F o ( S ) − u + diag ˜ ♮ N ) . Here ˜ ♮ N is the following sequence of length N : ˜ ♮ N = ( ( N − , N − , . . . , , , . . . , − N + 1) , N : even , ( N − , N − , . . . , , , − , . . . , − N + 1) , N : odd . This can be generalized as follows: Theorem 7.3 (Wachi) . For any u ∈ C , the following element is central in U ( o ( S )) : C o N k ( u ) = X ≤ i < ···
Here e F o ( S ) is defined by e F o ( S ) = ( F o ( S ) + diag(0 , . . . , , , . . . , , N : even ,F o ( S ) + diag(0 , . . . , , , , . . . , , N : odd . Here the numbers of ’s and ’s are equal to [ N/ . In Section 8, we will express the eigenvalues of these central elements in terms of theSchur type functions associated with the central difference.7.3. Similarly, we can construct central elements in the universal enveloping algebra ofthe symplectic Lie algebra. However, these are expressed in terms of permanents (not interms of determinants).Let J ∈ Mat N ( C ) be a non-singular alternating matrix of size N . We can realize thesymplectic Lie group as the isometry group with respect to the bilinear form determinedby J : Sp ( J ) = { g ∈ GL N | t gJ g = J } . The corresponding Lie algebra is expressed as sp ( J ) = { Z ∈ gl N | t ZJ + J Z = 0 } . As generators of this sp ( J ), we can take F sp ( J ) ij = E ij − J − E ji J , where E ij is the standardbasis of gl N . We introduce the N × N matrix F sp ( J ) whose ( i, j )th entry is this generator: F sp ( J ) = ( F sp ( J ) ij ) ≤ i,j ≤ N . We regard this matrix as an element of Mat N ( U ( sp ( J ))).We consider the split realization of the symplectic Lie algebra. Namely we consider thecase of J = J = ... − ... − . In this case, we can take the following triangular decomposition: sp ( J ) = n − ⊕ h ⊕ n + . Here n − , h , and n + are the subalgebras of sp ( J ) spanned by the elements F sp ( J ) ij suchthat i > j , i = j , and i < j , respectively. Namely, the entries in the lower triangular part,in the diagonal part, and in the upper triangular part of the matrix E gl N belong to n − , h , and n + , respectively. In this case, we can construct central elements of the universalenveloping algebra using the column-permanent [I6]: Theorem 7.4 (Itoh) . For any u ∈ C , the following element is central in U ( sp ( J )) : D sp N k ( u ) = X ≤ i ≤···≤ i k ≤ N I ! per( e F sp ( J ) I + u I − I diag( k − , k − , . . . , − k )) . Here we put e F sp ( J ) = F sp ( J ) − diag(0 , . . . , , , . . . , , where the numbers of ’s and ’s are equal to N/ . We can easily calculate the eigenvalue of this central element noting the triangulardecomposition of sp ( J ) and the definition of the column-permanent. In Section 8, we willexpress them in terms of the Schur type functions associated with the central difference. Remark. We note some previous results:(1) We can also express C o N k (0) and D sp N k (0) in terms of the symmetrized determinantand the symmetrized permanent ([W], [I5], [I6]). Capelli type identities in terms of thesesymmetrized determinant and permanent are also given in [I3] and [I4].(2) Analogues of the quantum immanants in U ( o N ) and U ( sp N ) are studied in [OO2]. Wewill see in Section 8 that these can be regarded as a generalization of C o N k (0) and D sp N k (0).8. The eigenvalues of Capelli type elements Finally we consider concrete examples of Schur type functions associated with somedifferences. When Q is the forward difference, the associated Schur type functions areequal to the factorial Schur functions. By the shift of variables, these are transformedinto the shifted Schur functions, and these are useful to express the eigenvalues of somecentral elements of the universal enveloping algebra of the general linear Lie algebra.When Q is the central difference, the associated Schur functions are useful to express theeigenvalues of the central elements of the universal enveloping algebras of the orthogonaland symplectic Lie algebras listed in the previous section.8.1. Let us consider the case that Q is equal to the forward difference ∆ + . In thiscase, p ∆ + n ( x ) and p ∗ ∆ + n ( x ) are expressed as p ∆ + n ( x ) = x n and p ∗ ∆ + n ( x ) = ( x − n . Thecorresponding symmetric functions e ∆ + k and h ∆ + k are explicitly expressed as follows ( e ∗ ∆ + k and h ∗ ∆ + k are also given by considering the shift of variables). This expression is essentiallyequivalent with Corollary 11.3 in [OO1]. Theorem 8.1. We have e ∆ + k ( x , . . . , x N )= X ≤ i < ···
The first relation is obtained from the relation e N ( x − u, . . . , x N − u ) = ( x − u ) · · · ( x N − u ) . It suffices to apply Q u repeatedly and use the Leibnitz rule for the forward difference:∆ + ( f ( x ) g ( x )) = ∆ + f ( x ) g ( x + 1) + f ( x )∆ + g ( x ). The second relation is deduced byinduction on N (use Theorems 5.2 and 5.3). (cid:3) Using these symmetric functions, we can describe the eigenvalues of C gl N k ( u ) and D gl N k ( u )defined in the previous section as follows: Theorem 8.2. For the representation π gl N λ of gl N determined by the partition λ =( λ , . . . , λ N ) , we have π gl N λ ( C gl N k ( u )) = e ∆ + k ( l , . . . , l N ; u ) , π gl N λ ( D gl N k ( u )) = h ∆ + k ( l , . . . , l N ; u ) . Here we put l i = λ i + N − i . CHUR FUNCTIONS FOR POLYNOMIAL SEQUENCES OF BINOMIAL TYPE 21 We can deduce this by the procedure outlined in Section 7 using the triangular decom-position of the general linear Lie algebra and the definitions of the column-determinantand the column-permanent.Theorem 8.2 is essentially included in the following description of the eigenvalues of thequantum immanants in terms of the factorial (shifted) Schur functions due to [OO1]: π gl N λ ( S µ ) = s ∆ + µ ( l , . . . , l N ) . Indeed, when u = 0, Theorem 8.2 is a special case of this relation as seen from (7.3).Moreover, the case of u = 0 can be also deduced from this by considering the algebraautomorphisms E ij E ij + uδ ij and E ij E ij − uδ ij on U ( gl N ).8.2. Next, let us consider the case of the central difference (namely Q = ∆ ). In thiscase, p ∆ n ( x ) and p ∗ ∆ n ( x ) are expressed as p ∆ n ( x ) = x · x n − and p ∗ ∆ n ( x ) = x n . Moreover e ∆ k and h ∗ ∆ k are expressed as follows: Theorem 8.3. We have e ∆ k ( x , . . . , x N )= X ≤ i < ···
For the irreducible representation π o N λ of o N determined by the partition λ = ( λ , . . . , λ [ n ] ) , we have π o N λ ( C o N k ( u )) = e ∆ k ( l , . . . , l N ; u ) . Here we define l , . . . , l N as follows. First, for ≤ i ≤ n , we put l i = λ i + n − i ( that is,we consider the ρ -shift ) . Next, we put l n +1 = − l n , . . . , l N = − l when N is even, and weput l n † = 0 , l n † +1 = − l n † − , . . . , l N = − l with n † = N +12 when N is odd. Similarly we have the following relation for D sp N k ( u ): Theorem 8.5. For the irreducible representation π sp N λ of sp N determined by the partition λ = ( λ , . . . , λ n ) , we have π sp N λ ( D sp N k ( u )) = h ∗ ∆ k ( l , . . . , l N ; u ) . Here l i is defined as follows: we put l i = λ i + n + 1 − i for ≤ i ≤ n ( that is, we considerthe ρ -shift ) , and put l n +1 = − l n , . . . , l N = − l for n + 1 ≤ i ≤ N . Remark. Factorial powers and differences were key tools in the study of Capelli typeelements, and various relations for them have been given ([I1–6], [IU], and [U1–5]). Theformulas in Sections 4 and 5 of this article can be regarded as natural generalizations ofthese relations.8.3. The functions e ∆ k and h ∗ ∆ k are related with analogues of the shifted Schur functionsdue to Okounkov and Olshanski. Moreover, from these relations and the calculation ofthe eigenvalues in Theorems 8.4 and 8.5, we see that C o N k (0) and D sp N k (0) are equal tospecial cases of analogues of the quantum immanants. Let us see these relations.In [OO2], Okounkov and Olshanski introduced analogues of the quantum immanantsin the universal enveloping algebras of the simple Lie algebras of types B, C, and D, anddenoted these elements by T µ . Let us write these as T Bµ , T Cµ , and T Dµ , when we need toindicate the type of the Lie algebra. Noting that the eigenvalues of these central elementsare polynomials in l , . . . , l n , Okounkov and Olshanski also defined a class of functions t ∗ µ corresponding to the types B, C, and D by the relation π λ ( T µ ) = t ∗ µ ( λ , . . . , λ n ) , (8.1)where π λ is the irreducible representation of the Lie algebra determined by the partition λ . We can regard t ∗ µ as an analogue of the shifted Schur functions, and this is expressedas follows (Lemma-Definition 2.4 in [OO2]): t ∗ Bµ ( λ , . . . , λ n ) = s µ (( λ + n − ) , ( λ + n − ) , . . . , ( λ n + ) | ( ) , ( ) , ( ) , . . . ) ,t ∗ Cµ ( λ , . . . , λ n ) = s µ (( λ + n ) , ( λ + n − , . . . , ( λ n + 1) | , , , . . . ) ,t ∗ Dµ ( λ , . . . , λ n ) = s µ (( λ + n − , ( λ + n − , . . . , ( λ n + 0) | , , , . . . ) . Here, the superscripts B , C , and D indicate the type of the corresponding Lie algebra.Moreover, the right hand sides are the “generalized factorial Schur functions” (see [OO2]for the definition). CHUR FUNCTIONS FOR POLYNOMIAL SEQUENCES OF BINOMIAL TYPE 23 The connection between these functions and e ∆ k , e ∗ ∆ k , h ∆ k , h ∗ ∆ k is seen from thefollowing relations: e ∆ k ( l , . . . , l n , − l n , . . . , − l ) = s (1 k ) ( l , . . . , l n | , , . . . ) ,e ∗ ∆ k ( l , . . . , l n , − l n , . . . , − l ) = s (1 k ) ( l , . . . , l n | ( ) , ( ) , . . . ) ,e ∗ ∆ k ( l , . . . , l n , , − l n , . . . , − l ) = s (1 k ) ( l , . . . , l n | , , . . . ) ,h ∗ ∆ k ( l , . . . , l n , − l n , . . . , − l ) = s ( k ) ( l , . . . , l n | , , . . . ) ,h ∆ k ( l , . . . , l n , − l n , . . . , − l ) = s ( k ) ( l , . . . , l n | ( ) , ( ) , . . . ) . These relations themselves follow from Theorems 5.1–5.3. The left hand sides of thesecond and fourth equalities are also equal to e ∆ k ( l , . . . , l n , , − l n , . . . , − l ) , h ∆ k ( l , . . . , l n , , − l n , . . . , − l ) , respectively (recall Proposition 2.3).From these relations, we can rewrite Theorem 8.4 in the case of u = 0 as π o n λ ( C o n k (0)) = t ∗ D (1 k ) ( λ , . . . , λ n ) , π o n +1 λ ( C o n +1 k (0)) = t ∗ B (1 k ) ( λ , . . . , λ n ) . Similarly, Theorem 8.5 in the case of u = 0 can be rewritten as π sp n λ ( D sp n k (0)) = t ∗ C ( k ) ( λ , . . . , λ n ) . Combining these with (8.1), we have the following theorem: Theorem 8.6. We have C o n k (0) = T D (1 k ) , C o n +1 k (0) = T B (1 k ) , D sp N k (0) = T C ( k ) . Indeed, the eigenvalues of both sides are equal.This theorem means that we can express T D (1 k ) , T B (1 k ) , and T C ( k ) in terms of the column-determinant and the column-permanent. However, an explicit descripsion of T µ in termsof a certain noncommutative determinant type function is not given for general µ . Thusthe three elements in Theorem 8.6 are lucky exceptions. In [MN], these three elementsare studied in more detail, and Capelli type identities for the dual pair ( O M , Sp N ) aregiven.In Theorems 8.4 and 8.5, we expressed the eigenvalues of C o N k ( u ) and D sp N k ( u ) naturallyin terms of the functions e ∆ k and h ∗ ∆ k for general u . This is an advantage of these functionsover the function t ∗ µ . Indeed, we cannot describe these eigenvalues in terms of t ∗ Bµ , t ∗ Cµ ,and t ∗ Dµ so simply. However, it should be noted that, even if u = 0, we can express theeigenvalue of C o N k ( u ) as a linear combination of t ∗ D (1 ) , t ∗ D (1 ) , . . . , t ∗ D (1 k ) or t ∗ B (1 ) , t ∗ B (1 ) , . . . , t ∗ B (1 k ) using Theorem 4.1. Similarly, the eigenvalue of D sp N k ( u ) can be expressed as a linearcombination of t ∗ C (0) , t ∗ C (1) , . . . , t ∗ C ( k ) by using Theorem 4.2.In the various relations in this article, there was a mysterious duality in the exchanges s ↔ s ∗ and λ ↔ λ ′ . It is also mysterious that the functions e and h ∗ played moreimportant roles than e ∗ and h (note that we can rewrite Theorem 8.2 in terms of h ∗ replacing u by u − U ( o N ) (respectively, U ( sp N )) expressed in terms of the column-permanent (respectively, the column-determinant) are not known. The author hopesthat the theoretical background of these phenomena will become transparent, and the Schur type functions in this article will be useful to study the analogues of the quantumimmanants in U ( o N ) and U ( sp N ) (especially to give their explicit description). References [BL] L. C. Biedenharn and J. D. Louck, A new class of symmetric polynomials defined in terms oftableaux , Advances in Appl. Math. (1989), 396–438.[Ca1] A. Capelli, ¨Uber die Zur¨uckf¨uhrung der Cayley’schen Operation Ω auf gew¨ohnliche Polar-Operationen , Math. Ann. (1887), 331–338.[Ca2] , Sur les op´erations dans la th´eorie des formes alg´ebriques , Math. Ann. (1890), 1–37.[CL] W. Y. C. Chen and J. D. Louck, The factorial Schur function , J. Math. Phys. (1993), 4144–4160.[HU] R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-freeactions , Math. Ann. (1991), 565–619.[I1] M. Itoh, Capelli elements for the orthogonal Lie algebras , J. Lie Theory (2000), 463–489.[I2] , A Cayley-Hamilton theorem for the skew Capelli elements , J. Algebra (2001), 740–761.[I3] , Capelli identities for the dual pair ( O M , Sp N ), Math. Z. (2004), 125–154.[I4] , Capelli identities for reductive dual pairs , Adv. Math. (2005), 345–397.[I5] , Two determinants in the universal enveloping algebras of the orthogonal Lie algebras , J.Algebra (2007), 479–506.[I6] , Two permanents in the universal enveloping algebras of the symplectic Lie algebras , toappear in Internat. J. Math.[IU] M. Itoh and T. Umeda, On central elements in the universal enveloping algebras of the orthogonalLie algebras , Compositio Math. (2001), 333–359.[M1] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford Science Publ.,1995.[M2] , Schur functions: themes and variations , in “S´eminaire Lotharingien de Combinatoire,”Publ. I.R.M.A. Strasbourg , 1992, pp. 5–39.[MN] A. Molev and M. Nazarov, Capelli identities for classical Lie algebras , Math. Ann. (1999),315–357.[MR] R. Mullin and G.-C. Rota, On the Foundations of Combinatorial Theory III: Theory of BinomialEnumeration , in “Graph Theory and Its Applications,” edited by Bernard Harris, Academic Press,New York, 1970, 167–213.[N] M. Nazarov, Quantum Berezinian and the classical Capelli identity , Lett. Math. Phys. (1991),123–131.[O] A. Okounkov, Quantum immanants and higher Capelli identities , Transform. Groups (1996),no. 1, 99–126.[OO1] A. Okounkov and G. Olshanski, Shifted Schur functions , Algebra i Analiz (1997), No. 2 (Rus-sian); English version in St. Petersburg Math. J. (1998), 239–300.[OO2] , Shifted Schur functions II. Binomial formula for characters of classical groups and ap-plications , in “A.A.Kirillov Seminar on Representation Theory,” Amer. Math. Soc. Translations(2) Vol. 181, (1998), pp. 245–271.[R] Steven Roman, The Umbral Calculus, Academic Press, New York, 1984; republication: DoverPublications, 2005.[RKO] G.-C. Rota, D. Kahaner, and A. Odlyzko, Finite Operator Calculus , Journal of MathematicalAnalysis and its Applications , no. 3 (1973), 684–760.[S] R.P. Stanley, Enumerative combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics,62. Cambridge University Press, Cambridge, 1999.[U1] T. Umeda, The Capelli identities, a century after , S¯ugaku (1994), 206–227, (in Japanese);English transl. in “Selected Papers on Harmonic Analysis, Groups, and Invariants,” AMS Trans-lations, Series 2, vol. 183 (1998), pp. 51–78, ed. by K. Nomizu.[U2] , Newton’s formula for gl n , Proc. Amer. Math. Soc. (1998), 3169–3175. CHUR FUNCTIONS FOR POLYNOMIAL SEQUENCES OF BINOMIAL TYPE 25 [U3] , On Turnbull identity for skew symmetric matrices , Proc. Edinburgh Math. Soc. (2) (2000), 379–393.[U4] , Application of Koszul complex to Wronski relations for U ( gl n ), Comment. Math. Helv. (2003), 663–680.[U5] , On the proof of the Capelli identities , Funkcialaj Ekvacioj (2008), 1–15.[W] A. Wachi, Central elements in the universal enveloping algebras for the split realization of theorthogonal Lie algebras , Lett. Math. Phys. (2006), 155–168. Department of Mathematics and Computer Science, Faculty of Science, KagoshimaUniversity, Kagoshima 890-0065, Japan E-mail address ::