Bilinear expansion of Schur functions in Schur Q -functions: a fermionic approach
aa r X i v : . [ m a t h - ph ] S e p Fermionic approach to bilinear expansions of Schurfunctions in Schur Q -functions J. Harnad , ∗ and A. Yu. Orlov † Centre de recherches math´ematiques, Universit´e de Montr´eal,C. P. 6128, succ. centre ville, Montr´eal, QC H3C 3J7 Canada Department of Mathematics and Statistics, Concordia University1455 de Maisonneuve Blvd. W. Montreal, QC H3G 1M8 Canada Institute of Oceanology, Nahimovskii Prospekt 36, Moscow 117997, Russia
Abstract
An identity is derived expressing Schur functions as sums over products of pairsof Schur Q -functions, generalizing previously known special cases. This is shown tofollow from their representations as vacuum expectation values (VEV’s) of productsof either charged or neutral fermionic creation and annihilation operators, Wick’stheorem and a factorization identity for VEV’s of products of two mutually anti-commuting sets of neutral fermionic operators. Fermionic methods are central to Sato’s construction of τ -functions for the KP infiniteintegrable hierarchy, as well as the the BKP hierarchy [3–8, 11]. In this work, we makeuse of the relations between charged and neutral fermionic operators to derive a bilinearidentity relating Schur functions [9], which are the basic building blocks for solutions ofthe KP hierarchy [6, 11], to Schur’s Q -functions, which play a similar rˆole with respect tothe BKP hierarchy [3, 12].An identity was derived in [1], expressing determinants of submatrices of skew matricesas sums over products of the Pfaffians of their principal minors. Geometrically, this may beinterpreted as a bilinear relation between the Pl¨ucker map , which embeds Grassmanniansof k -dimensional subspaces of a given vector space into the projectivization of the spaceof exterior k -forms, and the Cartan map [2], which embeds the Grassmannian of maximalisotropic subspaces with respect to a complex scalar product into the projectivization ofthe irreducible spinor modules. The result in [1] was based on Cartan’s construction ofbilinear forms on Clifford modules, with values in spaces of homogeneous exterior forms.The main result derived here is Theorem 5.1, Section 5), which may be viewed as afunction theoretic realization of this identity, with the determinant identified, through ∗ e-mail:[email protected] † e-mail:[email protected] Q -functions.Section 2 recalls the definition of creation and annihilation operators on fermionicFock space as linear generators of an infinite dimensional Clifford algebra. Two mutuallyanticommuting subalgebras generated by neutral fermions are defined, and a key fac-torization Lemma 2.3 given for vacuum state expectation values (VEV’s) of products oflinear elements. The representation of Schur functions and Schur Q-functions as VEV’sof products of creation and annihilation operators conjugated by elements of the infiniteabelian groups that generate KP and BKP flows is recalled in Section 3. Section 4 in-troduces the notion of polarizations associated to integer partitions, and the associatedproducts of neutral fermion operators determined by binary sequences. The main resultis Theorem 5.1, Section 5, which gives an expression for Schur functions, evaluated on theodd KP flow variables, as sums over products of pairs of Schur Q -functions, generalizingcertain previously known special cases [9] to arbitrary Schur functions. For a separable Hilbert space H , with orthonormal basis { e j } j ∈ Z , the correspondingfermionic Fock space F is defined as the semi-infinite wedge product space [6, 11]: F = Λ ∞ / H = M n ∈ Z F n , (2.1)with elements denoted as ket vectors | w i . The sector F n with fermionic charge n ∈ Z hasan orthonormal basis, denoted {| λ ; n i} , labelled by pairs ( λ, n ) of an integer partitition λ and the integer n , defined by: | λ ; n i := e l ∧ e l ∧ · · · , (2.2)where the infinite sequence of integers { l i } i ∈ N + , called particle positions , are related tothe parts ( λ ≥ λ · · · , λ ℓ ( λ ) , , . . . ) of λ by l i := λ i − i + n, i ∈ N + . (2.3)The length ℓ ( λ ) of the partition λ is the number of positive parts { λ i } i =1 ,...,ℓ ( λ ) and thefinite sequence is completed by adding an infinite sequence of 0’s following these. Its weight is | λ | := ℓ ( λ ) X i =1 λ i . (2.4)The particle positions { l i } i ∈ N + , form a strictly decreasing sequence that saturates, after ℓ ( λ ) terms, to all subsequent consecutively decreasing integers.2he algebra of fermionic operators on F form an irreducible representationΓ : C H + H ∗ ( Q ) → End( F )Γ : ξ Γ ξ (2.5)of the Clifford algebra C H + H ∗ ( Q ) on H + H ∗ corresponding to the (complex) scalar product Q defined by Q ( u + µ, v + ν ) := µ ( v ) + ν ( u ) , u, v ∈ H , µ, ν ∈ H ∗ . (2.6)These are realized as endomorphisms of F , with the linear elements acting by exteriorand interior multiplication:Γ v = v ∧ , Γ µ = i µ , v ∈ H , µ ∈ H ∗ . (2.7)Denoting the dual basis for H ∗ as { e j } j ∈ Z , with e j ( e k ) = δ jk , (2.8) C H + H ∗ ( Q ) is generated by the scalars and linear elements, with the representations of thebasis elements denoted ψ j := e j ∧ , ψ † j := i e j , j ∈ Z . (2.9)These are referred to as (charged) fermionic creation and annihilation operators, respec-tively, and satisfy the anticommutation relations:[ ψ j , ψ k ] + = [ ψ † j , ψ † k ] + = 0 , [ ψ j , ψ † k ] + = δ jk . (2.10)The vacuum element | n i in each charge sector F n is the basis element correspondingto the trivial partition λ = ∅ : | n i := |∅ ; n i = e n − ∧ e n − ∧ · · · . (2.11)Elements of the dual space F ∗ are denoted as bra vectors h w | , with the dual basis {h λ ; n |} for F ∗ n defined by the pairing h λ ; n | µ ; m i = δ λµ δ nm . (2.12)For KP τ -functions, we need only consider the n = 0 charge sector F , and generally dropthe charge n symbol, denoting the basis elements simply as | λ i := | λ ; 0 i . (2.13)For j > ψ − j and ψ † j − (resp. ψ †− j and ψ j − ) annihilate the right (resp. left) vacua: ψ − j | i = 0 , ψ † j − | i = 0 , ∀ j > , (2.14) h | ψ †− j = 0 , h | ψ j − = 0 , ∀ j > . (2.15)3eutral fermions φ + j and φ − j are defined [3] as φ + j := ψ j + ( − j ψ †− j √ , φ − j := i ψ j − ( − j ψ †− j √ , j ∈ Z (2.16)(where i = √− φ + j , φ − k ] + = 0 , [ φ + j , φ + k ] + = [ φ − j , φ − k ] + = ( − j δ j + k, . (2.17)In particular, ( φ +0 ) = ( φ − ) = . (2.18)Acting on the vacua | i and | i , we have φ + − j | i = φ −− j | i = φ + − j | i = φ −− j | i = 0 , ∀ j > , ∀ j > , (2.19) h | φ + j = h | φ − j = h | φ + j = h | φ − j = 0 , ∀ j > , (2.20) φ +0 | i = − iφ − | i = √ ψ | i = √ | i , (2.21) h | φ +0 = i h | φ − = √ h | ψ † = √ h | . (2.22)The pairwise expectation values are: h | φ + j φ + k | i = h | φ − j φ − k | i = ( − k δ j, − k if k > , δ j, if k = 0 , k < , (2.23) h | φ + j φ − k | i = −h | φ − j φ + k | i = i δ j, δ k, . (2.24) For an even number of fermionic operators ( w , . . . , w L ) that anticommute:[ w j , w k ] + = 0 , ≤ j, k ≤ L, (2.25)the matrix with elements h | w j w k | i is skew symmetric, and Wick’s theorem implies thatthe vacuum state expectation value h | w · · · w L | i of the product is given by the Pfaffian h | w · · · w L | i = Pf ( h | w j w k | i ) ≤ j,k ≤ L . (2.26)On the other hand, if the odd elements w , w , . . . are linear combinations of cre-ation operators { ψ j } j ∈ Z and the even ones w , w , . . . linear combinations of annihilationoperator { ψ † j } j ∈ Z , Wick’s theorem implies h | w · · · w L | i = det ( h | w j w k | i ) j =1 , ,... ; k =2 , ,... . (2.27)4 .3 Current components and a factorization lemma The positive current components of charged fermions, defined as J n = X i ∈ Z ψ i ψ † i + n , n ∈ N + , (2.28)commute amongst themselves [ J n , J m ] = 0 , n, m ∈ N + , (2.29)and generate the KP flows [6, 11].The neutral fermion current components J Bn and ˆ J Bn are defined as J B + n := X j ∈ Z ( − j +1 φ + j φ + − j − n , J B − n := X j ∈ Z ( − j +1 φ − j φ −− j − n , n ∈ N + . (2.30)The even components { J B +2 p , J B − p } all vanish, while the odd ones mutually commute:[ J B +2 p − , J B +2 q − ] = 0 , [ J B − p − , J B − q − ] = 0 , [ J B +2 p − , J B − q − ] = 0 , p, q ∈ N + , (2.31)and both generate BKP flows [3, 4, 6].By (2.14), the positive current components J n annihilate the vacuum | i : J n | i = 0 , n ∈ N + (2.32)and, by (2.19), the neutral current components J Bn and ˆ J Bn annihilate the vacua | i and | i : J B + n | i = 0 , J B − n | i = 0 , J B + n | i = 0 , J B − n | i , n ∈ N + . (2.33)It also follows from (2.28), (2.30) and ψ j = φ + j − iφ − j √ , ψ †− j = ( − j φ + j + iφ − j √ , (2.34)that: Lemma 2.1.
For odd n = 2 p − , J p − = J B +2 p − + J B − p − , p ∈ N + . (2.35)The following factorization property of VEV’s is proved in [6] and [12]. Lemma 2.2. If a + and a − are (finite or infinite) sums of monomials in the { φ + i } i =0 ofeven and odd degrees, respectively, and ˆ a + and ˆ a − are sums of monomials in the { φ − i } i =0 of even and odd degrees, respectively, then h | ( a + + φ +0 a − )(ˆ a + + φ − ˆ a − ) | i = h | ( a + + φ +0 a − )(ˆ a + + φ − ˆ a − ) | i = h | a + | ih | ˆ a + | i . (2.36)5s an immediate corollary, it follows that: Lemma 2.3 ( Factorization).
If ( u − , . . . , u + n ) and ( u − , . . . , u − m ) are linear combinationsof the operators { φ + i } i ∈ Z and { φ − i } i ∈ Z respectively, the VEV of their product can befactorized as: h | u +1 · · · u + n u − · · · u − m | i = h | u +1 · · · u + n | ih | u − · · · u − m | i if n and m are even0 if n and m have different parity2 i h | u +1 · · · u + n φ +0 | ih | u − · · · u − m φ − | i if n and m are odd . (2.37)In particular, for odd n and m , h | u +1 · · · u + n φ +0 | i and h | u − · · · u − m φ − | i vanish unless theterms in the products u +1 · · · u + n and u − · · · u − m that are linear in φ +0 and φ − , respectively,are nonzero. Q -functions via fermions. Let t = ( t , t , t , · · · ) denote the infinite sequence of KP flow parameters, and define theabelian group Γ + of KP flows { γ + ( t ) := e P ∞ i =1 t i Λ i } , where Λ ∈ End( H ) is the upwardshift element Λ( e i ) = e i − . (3.1)These act on F via the fermionic representationˆ γ + ( t ) := e P ∞ n =1 J n t n . (3.2)and, by (2.32), they stabilize the vacuum elementˆ γ + ( t ) | i = | i . (3.3)We define ψ j ( t ), ψ † j ( t ) to be the conjugates of ψ j and ψ † j by ˆ γ + ( t ). ψ j ( t ) := ˆ γ + ( t ) ψ j (ˆ γ + ( t )) − , ψ † j ( t ) := ˆ γ + ( t ) ψ † j (ˆ γ + ( t )) − . (3.4)It follows that these satisfy the same anticommutation relations (2.10) as { ψ j , ψ † j } j ∈ Z :[ ψ j ( t ) , ψ k ( t )] + = [ ψ † j ( t ) , ψ † k ( t )] + = 0 , [ ψ ( t ) j , ψ † k ( t )] + = δ jk . (3.5)Let ( α | β ) be the Frobenius notation for a partition λ , with α = ( α , . . . , α r ) and β = ( β , . . . , β r ) the “arm” and “leg” lengths, respectively, along the principal diagonal ofthe corresponding Young diagram, and r the Frobenius rank (i.e., the number of elementson the principal diagonal). Viewed as functions of the normalized power sum symmetricfunctions [9] in some auxiliary set of variables { x a } a ∈ N t i := i p i = i ∞ X a =1 x ia , (3.6)6e may express the corresponding Schur function s ( α | β ) ( t ) [9] as the following vacuumstate expectation values [6, 11] s ( α | β ) ( t ) = ( − P rj =1 β j ( − r ( r − h | ψ α ( t ) · · · ψ α r ( t ) ψ †− β − ( t ) · · · ψ †− β r − ( t ) | i = ( − P rj =1 β j h | r Y j =1 (cid:16) ψ α j ( t ) ψ †− β j − ( t ) (cid:17) | i , (3.7)where, by (3.5), [ ψ α k ( t ) , ψ †− β j − ( t )] + = 0 , ∀ j, k ∈ Z . (3.8) Remark 3.1.
Giambelli identity.
Applying Wick’s theorem (2.27) to the right hand side of(3.7) gives the Giambelli identity [9] s ( α | β ) ( t ) = det (cid:16) ( − β j h | ψ α k ( t ) ψ †− β j − ( t ) | i (cid:17) ≤ j,k, ≤ r = det (cid:16) s ( α j | β k ) ( t ) (cid:17) ≤ j,k ≤ r , (3.9)expressing s ( α | β ) ( t ) as the determinant of the r × r matrix formed from the hook partition Schurfunctions for each pair of Frobenius indices. Q -functions Denoting the set of odd flow variables as t B = ( t , t , t , . . . ) , (3.10)these may be viewed as determining a subset { t ′ } of the t ’s , where t ′ := ( t , , t , , t , . . . ) . (3.11)Following [3, 4, 12], we define two mutually commuting abelian groups of BKP flowsΓ B + = { γ B + ( t B + ) } and Γ B − = { γ B − ( t B ) } , with Clifford representationsˆ γ B + ( t B ) := e P ∞ p =0 , J B +2 p − t p − , ˆ γ B − ( t B ) := e P ∞ q =1 J B − q − t q − , (3.12)and note that, by (2.33), these stabilize both the vacua | i and | i ˆ γ B + ( t B ) | i = | i , γ B + ( t B ) | i = | i , ˆ γ B − ( t B ) | i = | i , ˆ γ B − ( t B ) | i = | i . (3.13)Defining φ + j ( t B ) := ˆ γ B + ( t B ) φ + j (ˆ γ B + ( t B )) − , φ − j ( t B ) := ˆ γ B − ( t B ) φ − j ((ˆ γ B − ( t B )) − , (3.14)it follows that these satisfy the same anticommutation relations (2.17) as { φ + j , φ − k } j,k ∈ Z :[ φ + j ( t B ) , φ − k ( t B )] + = 0 , [ φ + j ( t B ) , φ + k ( t B )] + = [ φ − j ( t B ) , φ − k ( t B )] + = ( − j δ j + k, . (3.15)From Lemma 2.35, we have 7 emma 3.1. ψ j ( t ′ ) = φ + j ( t B ) − iφ − j ( t B ) √ , ψ †− j ( t ′ ) = ( − j φ + j ( t B ) + iφ − j ( t B ) √ .. (3.16) Remark 3.2.
It follows from eq. (2.33) that (2.19) - (2.21) are still valid if φ +0 and φ − arereplaced by φ +0 ( t B ) and φ − ( t B ), so we have φ +0 ( t B ) | i = φ +0 | i = | i , φ − ( t B ) | i = φ − | i = i | i . (3.17) From (3.16), it follows that (3.7) may equivalently be expressed as:
Lemma 3.2. s ( α | β ) ( t ′ ) = ( − r ( r +1) − r h | (cid:0) φ + α ( t B ) − iφ − α ( t B ) (cid:1) · · · (cid:0) φ + α r ( t B ) − iφ − α r ( t B ) (cid:1) × (cid:0) φ + β +1 ( t B ) + iφ − β +1 ( t B ) (cid:1) · · · (cid:0) φ + β r +1 ( t B ) + iφ − β r +1 ( t B ) (cid:1) | i . (3.18)To define Schur’s Q -functions Q α we begin, following [9], by defining an infinite skewsymmetric matrix ( Q ij ) i,j ∈ N , whose entries are symmetric functions of the infinite sequenceof indeterminates x = ( x , x , . . . ), via the following formula: Q ij ( x ) := ( q i ( x ) q j ( x ) + 2 P jk =1 ( − k q i + k ( x ) q j − k ( x ) if ( i, j ) = (0 , , i, j ) = (0 , , (3.19)where the q i ( x )’s are defined by the generating function: ∞ Y i =1 zx i − zx i = ∞ X i =0 z i q i ( x ) . (3.20)In particular Q ( j, ( x ) = − Q (0 ,j ) ( x ) = q j ( x ) for j ≥ . (3.21)For a strict partition α of even cardinality r (including a possible zero part α r = 0),let M α ( x ) denote the r × r skew symmetric matrix with entries( M α ( x )) ij := Q α i α j ( x ) , ≤ i, j ≤ r. (3.22)The Schur Q -function is defined as its Pfaffian [9] Q α ( x ) := Pf( M α ( x )) (3.23)and, for completeness, Q ∅ := 1 . (3.24)Equivalently, these may be viewed as functions q j ( t B ), Q ij ( t B ) of the odd (normal-ized) power sum symmetric functions t B = ( t , t , . . . ) t i − := 12 i − p i − ( x ) = 12 i − ∞ X a =1 x i − a , i = 1 , , . . . . (3.25)We then have the fermionic VEV formulae [3, 10, 12]: Q α (cid:0) t B (cid:1) = 2 r h | φ + α ( t B ) · · · φ + α r ( t B ) | i , (3.26)= 2 r h | φ − α ( t B ) · · · φ − α r ( t B ) | i , (3.27)which follow from the Pfaffian form (2.26) of Wick’s theorem.8 .3 Examples: “doubles”, hook partitions and an r = 2 case Consider a set of Frobenius indices α = ( α , . . . , α r ) , (3.28)with α i +1 > α i , α r ≥ α r = 0 allowed as a part). Following [9],let DP denote the set of strict partitions (with all parts ≥ α we definethe strict partition I ( α ) = ( I ( α ) , . . . , I r ( α )) ∈ DP (3.29)whose parts are obtained form the α i ’s by shifting upward by 1: I i ( α ) := α i + 1 . (3.30)If a partition λ is related to a strict partition I = ( I , . . . I r ) in such a way that itsFrobenius indices are λ = ( I , . . . I r | I − , . . . I r − , (3.31)it is called [9] the double of I , and denoted λ = D ( I ). Example 3.1.
It is known (see [9], Chapter III, Section 8, example 10 (b)) that s D ( α ) ( t ′ ) = ( − r (cid:0) Q α (cid:0) t B (cid:1)(cid:1) if r is even , − r (cid:0) Q ( α, (cid:0) t B (cid:1)(cid:1) if r is odd , (3.32)where D ( α ) is the double of α . To prove this using fermionic VEV’s, we use:( − β j +1 ψ α j ( t ′ ) ψ †− β j − ( t ′ ) = − (cid:16) φ + α j ( t B ) − iφ − α j ( t B ) (cid:17) (cid:16) φ + β j +1 ( t B ) + iφ − β j +1 ( t B ) (cid:17) = 12 (cid:16) φ + β j +1 ( t B ) φ + α j ( t B ) + φ − β j +1 ( t B ) φ − α j ( t B ) (cid:17) + i (cid:16) φ − β j +1 ( t B ) φ + α j ( t B ) − φ + β j +1 ( t B ) φ − α j ( t B ) (cid:17) , (3.33)so for α j = β j + 1, we have( − α j ψ α j ( t ′ ) ψ †− α j ( t ′ ) = − iφ + α j ( t B ) φ − α j ( t B ) . (3.34)From eqs. (3.7) and (3.34) it follows that s D ( α ) ( t ′ ) = ( − i ) r h | r Y m =1 φ + α j ( t B ) φ − α j ( t B ) ! | i = ( i ) r ( − r ( r +1) h | (cid:0) φ + α ( t B ) · · · φ + α r ( t B ) φ − α ( t B ) · · · φ − α r ( t B ) (cid:1) | i . (3.35)Applying Lemma 2.3, eqs. (3.26) and (3.27) gives (3.32).9 xample 3.2. It is also known (see [9], Chapter III, Section 8, example 10 (a, iv) ) that s ( j | k ) ( t ′ ) = (cid:0) q j ( t B ) q k +1 ( t B ) − Q j,k +1 ( t B ) (cid:1) = (cid:0) Q ( j, ( t B ) Q ( k +1 , ( t B ) − Q j,k +1 ( t B ) Q ∅ ( t B ) (cid:1) . (3.36)To derive this fermionically, we apply (3.7), which gives: s ( j | k ) ( t ′ ) = − h | ( φ + j ( t B ) − iφ − j ( t B ))( φ + k +1 ( t B ) + iφ − k +1 ( t B ) | i = h | φ + k +1 ( t B ) φ + j ( t B ) | i + h | φ − k +1 ( t B ) φ − j ( t B ) | i + i h | φ − k +1 ( t B ) φ + j | i − i h | φ + k +1 φ − j ( t B ) | i = h | φ + k +1 ( t B ) φ + j ( t B ) | i + h | φ − k +1 ( t B ) φ − j ( t B ) | i + h | φ − k +1 ( t B ) φ − ( t B ) | ih | φ + j ( t B ) φ +0 ( t B ) | i + h | φ + k +1 ( t B ) φ +0 ( t B ) | ih | φ − j ( t B ) φ − ( t B ) | i = (cid:0) Q ( j, ( t B ) Q ( k +1 , ( t B ) − Q j,k +1 ( t B ) Q ∅ ( t B ) (cid:1) , (3.37)where Lemma 2.3 has been used in the third equality, and eqs. (3.17), (3.26) and (3.27)in the last. Example 3.3.
Consider a partition λ = ( α , α | β , α −
1) of Frobenius rank r = 2 inwhich α > β + 1 > α and β = α −
1, so s = 1. Expanding the RHS of eq. (3.18) forthis case, collecting together the four types of terms: h | φ + α ( t B ) φ + α ( t B ) φ − β +1 ( t B ) φ − α ( t B ) | i , h | φ + α ( t B ) φ − α ( t B ) φ − β +1 ( t B ) φ − α ( t B ) | i , h | φ + α ( t B ) φ + α ( t B ) φ − β +1 ( t B ) φ + α ( t B ) | i , h | φ + α ( t B ) φ − α ( t B ) φ − β +1 ( t B ) φ + α ( t B ) | i , (3.38)applying Lemma 2.3 and using eqs. (3.26) and (3.27) gives s ( α ,α | β ,α − ( t ′ ) = 14 Q ( α ,α ) ( t B ) Q ( β +1 ,α ) ( t B ) − Q ( α ,β +1 ,α , ( t B ) Q ( α , ( t B ) . (3.39)Following some preparatory definitions in Section 4, Theorem 5.1, Section 5 providesa generalization of these identities expressing s ( α,β ) ( t ′ ), for arbitrary partitions ( α | β ), asbilinear sums over products of Schur Q -functions. Let ( α | β ) be a partition of Frobenius rank r . Denote the union and intersection of α with I ( β ) as S := α ∩ I ( β ) , T := α ∪ I ( β ) , (4.1)and their cardinalities (or lengths, when viewed as strict partitions) as s := S ) = ℓ ( S ) , t := T ) = ℓ ( T ) = 2 r − s. (4.2)10 efinition 4.1. Polarizations. For any partition ( α | β ) a polarization is a pair of strictpartitions µ := ( µ + , µ − ) (4.3)(with 0 ’s allowed as parts), such that the following conditions are satisfied µ + ∩ µ − = S = α ∩ I ( β ) , µ + ∪ µ − = T = α ∪ I ( β ) . (4.4)Let P ( α, β ) denote the set of all polarizations corresponding to a partition ( α | β ) . Thefollowing Lemma shows that the cardinality of P ( α, β ) is 2 r − s . Lemma 4.1.
The number of distinct polarizations ( µ + , µ − ) corresponding to a pair ofstrict partitions S ⊂ T , with T of cardinality 2 r − s and S of cardinality s is 2 r − s . Proof.
The total number of elements in T is 2 r − s , and the number of these that arein S , and hence in both µ + and µ − is s . The remaining 2( r − s ) elements are either inone or the other of the two strict partitions µ ± , but not both, and hence there are 2 r − s ) distinct ways to select the polarization ( µ + , µ − ).Let m + ( µ ) := µ + ) , m − ( µ ) = µ − ) , (4.5)denote the cardinalities of µ + and µ − . Their sum is m + ( µ ) + m − ( µ ) = 2 r, (4.6)and therefore, they are either both even or both odd. Denoting the cardinalities of theintersections α ∩ µ − and I ( β ) ∩ µ − π ( µ ) := α ∩ µ − ) , ˜ π ( µ ) := I ( β ) ∩ µ − ) , (4.7)it follows that π ( µ ) + ˜ π ( µ ) = m − ( µ ) + s. (4.8) Definition 4.2. Binary markings.
For any integer j between 0 and 2 r −
1, we havethe associated binary sequence ǫ ( j ) := ( ǫ ( j ) , . . . , ǫ r ( j )) , (4.9)where ǫ k ( j ) = + if the k th element in the binary representation of j is 0 and ǫ k ( j ) = − ifthe k th element is 1. We call the product φ ( j ) ( α, β ) := φ ǫ ( j ) α · · · φ ǫ r ( j ) α r φ ǫ r +1 ( j ) β +1 · · · φ ǫ r ( j ) β r +1 (4.10)the j th binary marking of the sequence ( α, I ( β )).11f any two factors in (4.10) coincide, it vanishes. The number of these is 2 r − s (2 s − φ ( j ) ( α, β )’s is 2 r − s . For the latter, there are s pairs( m, n ) of type ( φ + α m , φ − β n +1 ) and ( φ − α m , φ + β n +1 ) where α m = β n + 1 ∈ S . By reordering theproduct (4.10) so that all the φ + ’s appear to the left, and the φ − ’s appear to the right,with all the subscripts in each group in decreasing order, every binary sequence ǫ ( j ) forwhich φ ( j ) ( α, β ) = 0 determines a unique polarization µ ( j ) = ( µ + ( j ) , µ − ( j )) such that φ ( j ) ( α, β ) =: ± φ + µ +1 · · · φ + µ + m +( µ ) φ − µ − · · · φ − µ + m − ( µ ) . (4.11)Of these, the polarization determined by ǫ (2 r −
1) is( µ + (2 r − , µ − (2 r − α, I ( β )) , (4.12)and this will be referred to as the canonical polarization .The 2 r − s binary sequences ǫ ( j ) for which φ ( j ) ( α, β ) = 0 may be divided into 2 r − s equivalence classes [ ǫ ( j )], each containing 2 s elements, for which the bilinear markings areequal, within a sign, and hence the polarization is the same:[ ǫ ( j )] = [ ǫ (˜ j )] if and only if µ ( j ) = µ (˜ j ) . (4.13)There is thus a bijection between the set P ( α, β ) of polarizations and the set of equivalenceclasses { [ ǫ ( j )] } of binary sequences which define (within a sign) the same nonvanishingbinary markings φ ( j ) ( α, β ). The 2 s elements of each equivalence class [ ǫ ( j )] are related byinterchanging any number of the s pairs ( m, n ) of type( φ + α m , φ − β n +1 ) ↔ ( φ − α m , φ + β n +1 ) , (4.14)where α m = β n + 1 ∈ S .The set of all α m ’s and β n + 1’s appearing in (4.10) with a + superscript, written indecreasing order, is the strict partition µ + ( j ) forming the first part of the polarization µ ( j ) and the set of all α m ’s and β n + 1’s in tappearing with a − superscript, also writtenin decreasing order, is the second part µ − ( j ).In every equivalence class ǫ ( j ), there is a unique element ǫ ( j ), for which ǫ α m ( j ) = + , ∀ α m ∈ S. (4.15)which will be referred to as the canonical representative . Remark 4.1.
There is one canonical representative ǫ ( j ) in each equivalence class [ ǫ ( j )],so the number of these is equal to the number 2 r − s of (nontrivial) equivalence classes,and they are in bijective correspondence with the polarizations µ ∈ P ( α, β ).Let σ ( j ) denote the parity of the number of α m ’s in the binary marking (4.10) thatare in S and appear with a superscript − , which equals the number of exchanges (4.14)of elements in the product (4.10) defining φ ( j ) ( α, β ) needed to convert it to φ ( j ) ( α, β ) φ ( j ) ( α, β ) = σ ( j ) φ ( j ) ( α, β ) , (4.16)In particular, σ ( j ) = 1. 12 efinition 4.3. Polarization sign. We define the sign of the polarization µ = µ ( j ),denoted sgn( µ ) by φ ( j ) ( α, β ) =: sgn( µ ) φ + µ +1 · · · φ + µ + m +( µ ) φ − µ − · · · φ − µ + m − ( µ ) . (4.17)It follows that φ ( j ) ( α, β ) = σ ( j ) sgn( µ ) φ + µ +1 · · · φ + µ + m +( µ ) φ − µ − · · · φ − µ + m − ( µ ) . (4.18) Example 4.1.
The partition λ = ((2 , , (1 , α = (2 , , I ( β ) = (2 , ,S = (2) , T = (2 , , , r = 2 , s = 1 . (4.19)The elements j are 2 , , = 4 distinct associated polarizations are µ (2) = µ (8) = (2 , , , (2)) , µ (3) = µ (9) = ((2 , , (2 , µ (6) = µ (12) = ((2 , , (2 , , µ (7) = µ (13) = ((2) , (2 , , . (4.20)The values of σ ( j ) and sgn( µ ( j )) for these are: σ (2) = + , σ (8) = − , sgn( µ (2)) = sgn( µ (8)) = + ,σ (3) = + , σ (9) = − , sgn( µ (3)) = sgn( µ (9)) = + ,σ (6) = + , σ (12) = − , sgn( µ (6)) = sgn( µ (12)) = − ,σ (7) = + , σ (13) = − , sgn( µ (7)) = sgn( µ (13)) = + . (4.21)The vanishing binary markings φ ( j ) ( α, β ) correspond to: j = 0 , , , , , , , Definition 4.4. Supplemented partitions. If µ is a strict partition of cardinality r (with 0 allowed as a part), define the associated supplemented partition ˆ µ to beˆ µ := ( µ, if r is even , ( µ, , if r is odd . (4.22)Note that ˆ µ is always of even cardinality, but not necessarily strict since, if r is odd,it may possibly have two 0 parts ( µ r = 0 ,
0) at the end. If m ± ( µ ) are the cardinalities of µ ± , we denote by ˆ m ± ( µ ) the cardinalities of ˆ µ ± . Q -functions Our main result expresses any Schur function, s ( α | β ) ( t ′ ), evaluated at t ′ , as a bilinear sumof products of Schur Q -functions. Theorem 5.1.
For any partition ( α | β ) = ( α , . . . , α r | β , . . . , β r ), we have s ( α | β ) ( t ′ ) = ( − r ( r +1)+ s r − s X µ ∈P ( α,β ) sgn( µ )( − π ( µ )+ 12 ˆ m − ( µ ) Q ˆ µ + ( t B ) Q ˆ µ − ( t B ) . (5.1)13 roof. Using Lemma 3.2 and expanding the product in (3.18) gives s α | β ) ( t ′ ) = ( − r ( r +1) r r − X j =0 ( − π ( µ ( j ))+ s i m − ( µ ( j )) σ ( j ) h | φ ( j ) ( t B )( α, β ) | i = ( − r ( r +1)+ s r − s X µ ∈P ( α,β ) ( − π ( µ ) i m − ( µ ) sgn( µ ) × h | φ + µ +1 ( t B ) · · · φ + µ + m +( µ ) ( t B ) φ − µ − ( t B ) · · · φ − µ + m − ( µ ) ( t B ) | i . (5.2)If m + ( µ ) and m − ( µ ) are both even, by Lemma 2.3 and eq. (3.26) we have h | φ + µ +1 ( t B ) · · · φ + µ + m +( µ ) ( t B ) φ − µ − ( t B ) · · · φ − µ + m − ( µ ) ( t B ) | i = h | φ + µ +1 ( t B ) · · · φ + µ + m +( µ ) ( t B ) | ih | φ − µ − ( t B ) · · · φ − µ + m − ( µ ) ( t B ) | i = 2 − r Q µ + ( ( t B ) Q µ − ( t B ) = 2 − r Q ˆ µ + ( t B ) Q ˆ µ − ( t B )) , (5.3)and i m − ( µ ) = ( − m − ( µ ) = ( −
12 ˆ m − ( µ ) . (5.4)If m + ( µ ) and m − ( µ ) are both odd, by Lemma 2.3 and eq. (3.27) we have h | φ + µ +1 ( t B ) · · · φ + µ + m +( µ ) ( t B ) φ − µ − ( t B ) · · · φ − µ + m − ( µ ) ( t B ) | i = 2 i h | φ + µ +1 ( t B ) · · · φ + µ + m +( µ ) ( t B ) φ − ( t B ) | ih | φ − µ − ( t B ) · · · φ − µ + m − ( µ ) ( t B ) φ +0 ( t B ) | i = 2 − r iQ ˆ µ + ( t B ) Q ˆ µ − ( t B )) , (5.5)and i m − ( µ )+1 = ( − m − ( µ )+1 = ( −
12 ˆ m − ( µ ) . (5.6)In both cases, substituting these in (5.2) gives (5.1). Remark 5.1.
Note that half the 2 r − s ) terms in the sum (5.1) are the same as the otherhalf (under the interchange ( µ + , µ − ) ↔ ( µ − , µ + )), leaving only 2 r − s − distinct terms(except for the case s = r , where there is just one). Example 5.1 (cf. Example 3.1) . Consider the case of a “double” D ( α ) = ( α , . . . , α r | α − , · · · , α r − , (5.7)for which s = r . The only binary sequence of type ǫ ( j ) (i.e. for which none of the elements α m corresponds to an upper index − ) is: ǫ (2 r −
1) = (0 , · · · | {z } r terms , , · · · | {z } r terms ) , (5.8)14hich gives the canonical polarization µ = µ (2 r −
1) = (( α , . . . , α ) , ( α , . . . , α )) . (5.9)For this case we havesgn( µ ) = +1 , π ( µ ) = r, ˆ m − ( µ ) = ( r if r is even r + 1 if r is odd . (5.10)Therefore ( − r ( r +1)+ s r − s sgn( µ )( − π ( µ )+ 12 ˆ m − ( µ ) = 12 r , (5.11)whether r is even or odd, and eq. (5.1) gives s D ( I ( α )) ( t ′ ) = ( − r (cid:0) Q α (cid:0) t B (cid:1)(cid:1) if r is even , − r (cid:0) Q ( α, (cid:0) t B (cid:1)(cid:1) if r is odd , (5.12)in agreement with eq. (3.32). Example 5.2 (cf. Example 3.2) . Consider a hook partition λ = ( α | β ) with α > β + 1.Then S = ∅ , ( r, s ) = (1 , µ :( µ + (0) , ( µ − (0)) = (( α , β + 1) , ( ∅ )) , ( µ + (1) , µ − (1)) = (( α ) , ( β + 1)) , ( µ + (2) , µ − (2)) = (( β + 1) , ( α )) , ( µ + (3) , µ − (3) = (( ∅ ) , ( α , β + 1)) . (5.13)Assuming α > β + 1 to fix the order in the notations and the correct sign factor, thevalues of sgn( µ ) , π ( µ ) and ˆ m − ( µ ) for each case is:sgn( µ (0)) = +1 , π ( µ (0) = 0 , ˆ m − ( µ (0)) = 0 , sgn( µ (1)) = +1 , π ( µ (1) = 0 , ˆ m − ( µ (1)) = 2 , sgn( µ (2)) = − , π ( µ (2) = 1 , ˆ m − ( µ (2)) = 2 , sgn( µ (3)) = +1 , φµ (3) = 1 , ˆ m − ( µ (3)) = 2 , (5.14)Substituting in eq. (5.1) gives s ( α | β ) ( t ′ ) = 12 Q ( α , ( t B ) Q ( β +1 , ( t B ) − Q ( α ,β +1) ( t B ) Q ∅ , (5.15)which is the same as (3.36). Example 5.3 (cf. Examples 3.2, 3.3, for r = 1 , . For r ≥
1, we have the followinggeneralization of Example 5.2. Assume the Frobenius indices satisfy β j + 1 = α k for allpairs except possibly one, say α > β + 1 > α , as in Example 3.3. Let λ = ( α , α , . . . , α r | β , α − , . . . , α r − , (5.16)15ith α > β + 1 > α if r >
1. Then s = r − { ( µ + ( i ) , µ − ( i )) } i =1 ,..., .( µ + (2 r − , µ − (2 r − α , α , . . . , α r ) , ( β + 1 , , α , . . . , α r )) , ( µ + (2 r − + 2 r − − , µ − (2 r − + 2 r − − β + 1 , α , . . . , α r ) , ( α , α , . . . , α r )) , ( µ + (2 r − + 2 r − , µ − (2 r − + 2 r − α , α , . . . , α r ) , ( α , β + 1 , α , . . . , α r )) , ( µ + (2 r − − , µ − (2 r − − α , β + 1 , α , . . . , α r , )( α , . . . , α r )) . (5.17)The corresponding values of sgn( µ ) , π ( u ) and ˆ m − ( µ ) are:sgn( µ (2 r − , π ( µ (2 r − r − , ˆ m − ( µ (2 r − r, sgn( µ (2 r − + 2 r − − − , π ( µ (2 r − + 2 r − − r, ˆ m − ( µ (2 r − + 2 r − − r, sgn( µ (2 r − + 2 r − − r − , π ( µ (2 r − + 2 r − r, ˆ m − ( µ (2 r − + 2 r − [ r + 1 , sgn( µ (2 r − − − r − , π ( µ (2 r − − r − , ˆ m − ( µ (2 r − − [ r − . (5.18)Substituting these in eq. (5.1) gives s ( α ,...,α r | β ,α − ,...,α r − ( t ′ ) = r (cid:0) Q ˆ µ + (2 r − ( t B ) Q ˆ µ − (2 r − ( t B ) − Q ˆ µ + (2 r − − ( t B ) Q ˆ µ − (2 r − − ( t B ) (cid:1) , (5.19)in agreement with (3.36) for r = 1 and (3.39) for r = 2. Acknowledgements.
The authors would like to thank Johan van de Leur and Ferenc Balogh forhelpful discussions. The work of J.H. was supported by the Natural Sciences and EngineeringResearch Council of Canada (NSERC); that of A. Yu. O. by RFBR grant 18-01-00273a.
References [1] F. Balogh, J. Harnad and J. Hurtubise, “Isotropic Grassmannians, Pl¨ucker and Cartanmaps”, arXiv:2007.03586[2] E. Cartan,
The Theory of Spinors , Dover Publications Inc, Mineola N.Y., 1981.[3] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, “Transformation groups for soliton equa-tions IV. A new hierarchy of soliton equations of KP type”,
Physica , 343-365 (1982).[4] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, “Transformation groups for soliton equa-tions”, In: Nonlinear integrable systems - classical theory and quantum theory , 39-120.World Scientifc (Singapore), eds. M. Jimbo and T. Miwa (1983).
5] J. Harnad and F. Balogh, “Tau functions and their applications”,
Monographs on Mathe-matical Physics , Cambridge University Press (in press, 2020).[6] M. Jimbo and T. Miwa, “Solitons and infinite-dimensional Lie algebras”,
Publ. Res. Inst.Math. Sci. , The Bispectral Problem, CRM Proceedings and Lecture Notes , Vol. , 159202. American Mathematical Society (1997).[8] J. van de Leur and A. Yu. Orlov, “Pfaffian and Determinantal Tau Functions”, Lett. Math.Phys.
Symmetric Functions and Hall Polynomials , Clarendon Press, Oxford,(1995).[10] J. J. C. Nimmo and A. Yu. Orlov, “A relationship between rational and multi-solitonsolutions of the BKP hierarchy”,
Glasgow Math. J. , , 149-168 (2005).[11] M. Sato. “Soliton equations as dynamical systems on infinite dimensional Grassmann man-ifold” Kokyuroku, RIMS
Infinite-Dimensional Lie Algebras and Groups , Adv. Ser. Math.Phys. (1989). World Sci. Publ., Teaneck, NJ.(1989). World Sci. Publ., Teaneck, NJ.