Commutation and normal ordering for operators on symmetric functions

S´eminaire Lotharingien de Combinatoire 80 (2019), Article B80d COMMUTATION AND NORMAL ORDERING FOR OPERATORS ON SYMMETRIC FUNCTIONS EMMANUEL BRIAND, PETER R. W. MCNAMARA, ROSA ORELLANA, AND MERCEDES ROSAS Dedicated to Ira Gessel on the occasion of his retirement Abstract. We study the commutation relations and normal ordering between families of operators on symmetric functions. These operators can be naturally defined by the operations of multiplication, Kronecker product, and their adjoints. As applications, we give a new proof of the skew Littlewood–Richardson rule and prove an identity about the Kronecker product with a skew Schur function. 1. Introduction Due to their connection to representation theory, Schubert calculus, and their beautiful combinatorial description, Schur functions are ubiquitous. For this reason, it is not surprising that identities involving Schur functions greatly improve the understanding of other subjects. There are two important products defined on symmetric functions: the ordinary and the Kronecker product. The well-known Littlewood–Richardson coefficients are the structure coefficients for the ordinary product, while the elusive Kronecker coefficients are the structure coefficients for the Kronecker product. They naturally define linear operators on symmetric functions. Let fbe a symmetric function. Then, define Ufto be the operator “multiplication by f”, and Kfbe “Kronecker multiplication by f.” Explicitly, the operators act on a symmetric function gby Ufpgq “ fg, Kfpgq “ f˚g. The Hall inner product allows us to define the adjoint of Uf, which is denoted by Df, and sometimes called the skewing operator. With respect to this inner product, Kfis self-adjoint. We will also consider another intriguing operator related to the Kronecker product, Kλ, defined on the Schur basis 1991 Mathematics Subject Classification. Primary 05E10, Secondary 05E05. Key words and phrases. Symmetric functions, Schur functions, normal ordering relations. 2 E. BRIAND, P.R.W. MCNAMARA, R. ORELLANA, AND M. ROSAS as follows: let λbe a partition and gany homogeneous symmetric function of degree n. Then Kλpgq “ spn´|λ|,λq˚g , where pn´ |λ|, λq:“ pn´ |λ|, λ1, λ2, . . .q. This sequence of integers is not always decreasing. To deal with this issue, we define the Schur function spn´|λ|,λqby means of the Jacobi–Trudi formula: spα1,α2,...,αNq“detphαi`j´iqi,j“1...N . This determinant coincides with the Schur function sαwhen αis weakly decreasing (i.e., is a partition) but makes sense even when αis not. Using linearity, we can define Kffor any symmetric function f. In this paper, we study identities involving four families of operators on the ring of symmetric functions, Sym:Uλ,Dλ,Kλ, and Kλ. We seek to address the following questions: what are the commutation relations between these operators? Given a word involving these operators, how can we put it in a normal form? Is this expression unique? Is it possible to express some of these operators in terms of the other ones? For a motivating example, let us look at the operators Up1qand Dp1q. They are well-known to satisfy the commutation relation Dp1qUp1q“Up1qDp1q`1. That is Leibniz’s rule for multiplication, when Up1qis multiplication by x, and Dp1q“B Bx. This identity is the defining relation for the algebra of Weyl, and the building identity for Stanley’s theory of differential posets [16]. Our first result is the following theorem that gives beautiful commutation relations for the operators. Theorem 1.1. For any partitions αand βwe have the following identities (where λ,τand νeach run over the set of all partitions). DβUα“ÿ λ Uα{λDβ{λ (1.1) UαDβ“ÿ λ p´1q|λ|Dβ{λ1Uα{λ (1.2) KβUα“ÿ λ Usβ{λ˚sαKλ (1.3) DαKβ“ÿ λ KλDsβ{λ˚sα (1.4) KβUα“ÿ τ,ν Upsβ{ν˚sτqsα{τKν (1.5) DαKβ“ÿ τ,ν KνDpsβ{ν˚sτqsα{τ (1.6) OPERATORS ON SYMMETRIC FUNCTIONS 3 The operators Kµand Kνcommute. To the best of our knowledge, the commutation relations for the pairs of operators pU, Kq,pU, Kq,pK, Dq, and pK,Dqare new results. Section 2.7 discusses the historical context of the results stated in Theorem 1.1. With this paper, we aim to show the important role that generating functions can play, providing unified tools to deal with symmetric functions. Towards this end, we associate a generating series with each of our operators. This allows us to obtain a uniform and elegant method to derive our results. More precisely, let Pbe any of the operators U,D,Kand K. We associate with Pa formal series of operators řλsλrAsPλ.We call this series the Schur generating series of P. The Schur generating series of Pdefines a linear map that sends any symmetric function gPSympXqto the expression řλsλrAsPλpgq.Then, we first study the existing relations between the Schur generating functions obtained in this way. For example, to derive our first expression we show that DσrBXsUσrAXs“σrABsUσrAXsDσrBXs. Unexpectedly, given the common nature of these results, and of their proofs, the different pairs of operators behave quite differently with regards to the question of uniqueness. In Proposition 2.6, we show that finite expansions with respect to the ordered pairs of operators pU, Dq,pD, Uq,pU, Kq, pK, Dq,pU, Kqand pK, Dqare unique. In contrast, observe that expansions with respect to pK, Uqor to pD, Kqare not unique. For instance we have the relation Kp2Up1“0, where pidenotes the power sum symmetric function řjxi j, which is equivalent to the relation K2U1“K1,1U1. Taking adjoints, we have the relation D1K2“D1K1,1. An open problem is to determine what happens for the remaining pairs of operators: are the corresponding expansions unique? The charming identity Kp1q“Up1qDp1q´1 describes a relation between the operators K, U, and D. In the second part of this work, we will see how it translates to one of the few known cases where it is possible to give a combinatorial description for a Kronecker product. Note that this identity shows that Kp1qcan be rewritten in terms of the much simpler operators Up1qand Dp1q. In Proposition 2.10 we vastly generalize this observation, and show that for any symmetric function fthe operator Kflies in the subalgebra of EndpSymqgenerated by the operators Ugand Dg. Dealing with naturally defined objects, like our families of operators, one is bound to recover some classical results. A testimony of the elegance of this approach is that both Foulkes’ (Eq. (2.22)) and Littlewood’s identities (Eq. (2.23)) can be easily derived from Theorem 2.11. However, a new identity of the same nature (Eq. (2.24)) relating the skewing and the Kronecker operators is also obtained, as described in the following table that shows 4 E. BRIAND, P.R.W. MCNAMARA, R. ORELLANA, AND M. ROSAS how the identities fit in relation to the different products and coproducts of symmetric functions. This is discussed in Section 2.7. Product Coproduct equivalent identities Ordinary, ¨Adjoint of the ordinary product (1.1) and (2.22) Kronecker, ˚Adjoint of the ordinary product (1.3) and (2.23) Ordinary, ¨Adjoint of the Kronecker product (1.4) and (2.24) Table 1.1. The three bialgebra structures on Sym and the corresponding identities Bx˝µ“µ˝ B∆pxq. The second part of our work is of combinatorial nature. Written for the reader with a combinatorial mind, it explores some applications of our results. The first application is a new proof of the skew Littlewood–Richardson rule, a combinatorial rule that gives the product of two skew Schur functions as a linear combination of skew Schur functions, based on counting Young tableaux (Theorem 3.1). This rule was conjectured in [1], and proved in [7]. Our proof relies on the normal ordering relation that decomposes the products UαDβas linear combinations of products of the form Dβ{λ1Uα{λ. It generalizes the algebraic proof given by Thomas Lam of the skew Pieri rule (a particular case of the skew Littlewood–Richardson rule) in the appendix of [1]. Indeed, Lam’s proof relies on the same normal ordering relation, in the special case of βhaving only one part. The second application exploits our normal ordering relation for the products K1Dλ. We extend the combinatorial rule for the expansion in the Schur basis of the Kronecker product of spn´1,1qwith a Schur function to the Kronecker product of spn´1,1qwith any skew Schur function (Theorem 3.2). This might be the only known instance of a combinatorial rule for the Kronecker product of a Schur function with a skew Schur function. We also give a combinatorial proof of this result. 2. Part I: Algebraic results In this section, we prove all six identities of Theorem 1.1. We begin with an overview of the method of proof. Let Pbe any of the families of operators U,D,Kand K. These operators act on Sym “SympXq, the symmetric functions in some alphabet X. Introduce an auxiliary alphabet A. The Schur generating series of Pis defined as ÿ λ sλrAsPλ. OPERATORS ON SYMMETRIC FUNCTIONS 5 This can be interpreted as the linear map that sends any symmetric function gPSympXqto the expression ÿ λ sλrAsPλpgq. For each of the four operators under consideration, the effect of the Schur generating function operator is described nicely by means of operations on alphabets (Lemma 2.2). The identities of Theorem 1.1 are derived at the level of generating series. The result is then recovered by extracting coefficients by means of the appropriate scalar product. 2.1. Preliminaries: operations on alphabets. Let X“ tx1, x2, . . .ube the underlying alphabet for the symmetric functions in Sym. Any infinite alphabet Agives rise to a copy SympAqof Sym. In this copy, the corresponding scalar product will be denoted by x | yA, and the element corresponding to fPSym, by frAs. Accordingly, the scalar product x | y of Sym and elements fPSym will be denoted sometimes by x | yXand frXs. If Aand Bare two alphabets, the tensor product SympAq b SympBqis endowed with the induced scalar product x | yA,B . Both the Kronecker product ˚and the adjoint Dfof the operator of multiplication by a symmetric function fwill be only considered with respect to Sym “SympXq. Given an algebra homomorphism Afrom Sym to some commutative algebra R, it will be convenient to write it as fÞÑ frAs(rather than fÞÑ Apfq) for any algebra homomorphism and consider it as a “specialization at the virtual alphabet A.” Since the power sum symmetric functions pk(kě1) generate Sym and are algebraically independent, the map (2.1) AÞÑ pp1rAs, p2rAs, . . .q is a bijection from the set of all algebra homomorphisms from Sym to R to the set of infinite sequences of elements from R. This set of sequences is endowed with its operations of component-wise sum and product, and multiplication by a scalar. The bijection (2.1) is used to lift these operations to the set of homomorphisms from Sym to R. This defines expressions like frA`Bsand frABs, where fis a symmetric function and Aand B are two “virtual alphabets,” and more general expressions frPpA, B, . . .qs, where PpA, B, . . .qis a polynomial in several virtual alphabets A,B. . . with coefficients in the base field. Note that, by definition, for any power sum pk (kě1), virtual alphabets Aand B, and scalar z, pkrA`Bs “ pkrAs ` pkrBs, pkrABs “ pkrAs ¨ pkrBs, pkrzAs “ z pkrAs. 6 E. BRIAND, P.R.W. MCNAMARA, R. ORELLANA, AND M. ROSAS In our calculations below, the homomorphism fÞÑ fr1swill appear: it is the specialization at x1“1, x2“0, x3“0 . . . , sending each pkto 1. The homomorphism fÞÑ frXsis just the identity of Sym. The homomorphism fÞÑ frXKs “ fK“Dfassociates to fthe adjoint of the operator Uf. Let σbe the generating series for the complete homogeneous symmetric functions hn, meaning σ“ř8 n“0hn, where h0“1. Recall from [9, I.§2] that we also have σ“exp ˜8 ÿ k“1 pk k¸“ÿ λ pλ zλ , where the last sum is over all partitions λ. We will make use of the following well–known identities. Lemma 2.1. Let Aand Bbe two alphabets, and fand gbe two symmetric functions. Then we have the following identities: σrA`Bs “ σrAsσrBs,(2.2) σrABs “ ÿ λ sλrAssλrBs,(Cauchy identity)(2.3) σr´ABs “ ÿ λ p´1q|λ|sλ1rAssλrBs,(2.4) DσrAXspfrXsq “ frX`As,(2.5) σrAXs ˚ frXs “ frAXs,(2.6) xfrAXs | grXs yX“ pf˚gqrAs,(2.7) xσrABs | grBs yB“grAs.(Reproducing Kernel)(2.8) 2.2. Generating series. Let Pbe any of the families of operators U,D, Kand K, and let Abe an auxiliary alphabet. We introduce the generating series for Pas ÿ λ sλrAsPλ. Using the linearity of fÞÑ Pfand the Cauchy identity (2.3), we may simplify this expression as follows: ÿ λ sλrAsPλ“ÿ λ sλrAsPsλrXs“PřλsλrAssλrXs“PσrAXs. Note that any operator Pλcan be recovered from the generating series by a coefficient extraction using a scalar product: Pλpfq “ @PσrAXspfrXsq ˇˇsλrAsDA. The generating series PσrAXsalso acts linearly on symmetric functions. The following lemma describes the effect of all four generating series UσrAXs, DσrAXs,KσrAXsand KσrAXs. OPERATORS ON SYMMETRIC FUNCTIONS 7 Lemma 2.2. Let frXsbe any symmetric function. Then UσrAXspfrXsq “ σrAXs ¨ frXs,(2.9) DσrAXspfrXsq “ frX`As,(2.10) KσrAXspfrXsq “ frAXs,(2.11) KσrAXspfrXsq “ σr´As ¨ frXpA`1qs.(2.12) Proof. Equation (2.9) follows by definition. Equation (2.10) is (2.5). Equation (2.11) is (2.6). We prove (2.12). For any symmetric functions fand g, Kfpgq “ Γ1f˚g, where Γ1is the vertex operator (2.13) Γ1“˜8 ÿ i“0 Upiq¸˜8 ÿ j“0 p´1qjDp1jq¸. We will make use of the following identity (see [13, §3]; for a combinatorial approach to this identity see [11]): (2.14) Γ1f“σrXsfrX´1s. Therefore, we have KσrAXspfq “ pΓ1σrAXsq ˚ frXs “σrX`ApX´1qs ˚ frXsby (2.2), “σr´As pσrXpA`1qs ˚ frXsq by (2.2), “σr´As ¨ frXpA`1qs by (2.6).  2.3. Operators Uand D.We now prove (1.1) and (1.2) of Theorem 1.1. The following lemma establishes the commutation relation between Uand D. Lemma 2.3. Let Aand Bbe two alphabets. We have DσrBXsUσrAXs“σrABsUσrAXsDσrBXs,(2.15) UσrAXsDσrBXs“σr´ABsDσrBXsUσrAXs.(2.16) Proof. Notice that (2.16) follows from (2.15) and the fact that σr´ABsis the inverse of σrABs(see (2.2)). We prove (2.15) using Lemmas 2.1 and 2.2. For any symmetric function frXs, we have DσrBXsUσrAXspfrXsq “ σrApX`BqsfrX`Bsby (2.10), “σrABsσrAXsfrX`Bsby (2.2), “σrABsσrAXsDσrBXspfrXsq by (2.10), “σrABsUσrAXsDσrBXspfrXsq. 8 E. BRIAND, P.R.W. MCNAMARA, R. ORELLANA, AND M. ROSAS Proof of (1.1).We use that, since Uand D:Sym ÑEndpSymqare algebra homomorphisms, for any symmetric function fwe may write that Uf“frUs and Df“frDs. In particular, for the generating series we have UσrAXs“ σrAUsand DσrBXs“σrBDs. In (2.15), the operator DβUαis the coefficient of sαrAssβrBsin the expansion in the Schur basis of σrABsUσrAXsDσrBXs, which is extracted by performing the scalar product with sαrAssβrBs. Thus, DβUα“@σrABsUσrAXsDσrBXsˇˇsαrAssβrBsDA,B “ÿ λ@sλrAssλrBsUσrAXsDσrBXsˇˇsαrAssβrBsDA,B by (2.3), “ÿ λ@σrAUsˇˇsα{λrAsDA@σrBDsˇˇsβ{λrBsDB “ÿ λ sα{λrUssβ{λrDs “ ÿ λ Uα{λDβ{λ. Identity (1.2) is derived from (2.16) analogously. 2.4. Operators Uand K.We now turn to the proof of (1.3). Identity (1.4) follows from (1.3) by taking adjoints. The following lemma gives the commutation relations for the generating series of Kand U. Lemma 2.4. Let Aand Bbe two alphabets. We have (2.17) KσrBXsUσrAXs“UσrABXsKσrBXs. Proof. For any symmetric function f, we have KσrBXsUσrAXspfrXsq “ KσrBXspσrAXsfrXsq “σrABXsfrBXsby (2.11), “UσrABXsKσrBXspfrXsq by (2.11).  Proof of (1.3).We get KβUαfrom (2.17) by extracting the coefficient of sβrBssαrAsin its expansion in terms of Schur functions: KβUα“@UσrABXsKσrBXsˇˇsβrBssαrAsDA,B “ÿ λ@UσrABXssλrBsˇˇsβrBssαrAsDA,B Kλ “ÿ λ@sαrBUsˇˇsβ{λrBsDBKλby (2.8), “ÿ λ psα˚sβ{λqrUsKλby (2.7), OPERATORS ON SYMMETRIC FUNCTIONS 9 “ÿ λ Usα˚sβ{λKλ. 2.5. Operators Uand K.We now proceed to proving (1.5). Identity (1.6) is deduced by taking adjoints. Again, we compute commutation relations for the generating series of the families of operators involved. Lemma 2.5. Let Aand Bbe two alphabets. We have (2.18) KσrBXsUσrAXs“UσrApB`1qXsKσrBXs. Proof. For any symmetric function f, we have KσrBXsUσrAXspfq “ KσrBXspσrAXsfrXsq “σr´BsσrAXpB`1qsfrXpB`1qs by (2.12), “σrAXpB`1qsKσrBXspfqby (2.12), “UσrApB`1qXsKσrBXspfq. Proof of (1.5).From (2.18) we extract the term KβUαby taking the scalar product with sαrAssβrBs. This yields KβUα“@UσrApB`1qXsKσrBXsˇˇsαrAssβrBsDA,B . Expanding the generating function KσrBXsin the scalar product and simplifying, we get KβUα“ÿ ν@UσrApB`1qXssνrBsˇˇsαrAssβrBsDA,B Kν, “ÿ ν@@UσrApB`1qXsˇˇsαrAsDAsνrBsˇˇsβrBsDBKν “ÿ ν@sαrpB`1qUssνrBsˇˇsβrBsDBKνby (2.8), “ÿ ν,τ @sτrBUsˇˇsβ{νrBsDBsα{τrUsKν “ÿ ν,τ psτ˚sβ{νqrUssα{τrUsKνby (2.7), “ÿ ν,τ Upsτ˚sβ{νqsα{τKν. In Section 3.5 we present as an application a combinatorial rule for the Kronecker product of any skew Schur function by spn´1,1q. Other interesting particular cases of (1.5) correspond to the cases when λ“ pkq(Kronecker 16 E. BRIAND, P.R.W. MCNAMARA, R. ORELLANA, AND M. ROSAS As mentioned in the introduction, Ira Gessel [4] established special cases of (1.1) and (1.2) when the Schur functions are indexed by one-row or onecolumn shapes. He showed that DnUm“ÿ i Um´iDn´i, UmDn“DnUm´Dn´1Um´1, Dp1nqUm“UmDp1nq`Um´1Dp1n´1q. Since Kβand Kβare self-adjoint, and Uαand Dαare adjoints of each other, (1.4) and (1.6) are obtained from (1.3) and (1.5), respectively, by taking adjoints. An interesting and elegant way of stating some of the results of Theorem 1.1 is in terms of commutators. Corollary 2.12. For any two partitions αand β, we have rDβ, Uαs “ ÿ λ‰H Uα{λDβ{λ“ÿ λ‰H p´1q|λ|´1Dβ{λ1Uα{λ, rKβ, Uαs “ ÿ pτ,νq‰pH,βq Upsβ{ν˚sτqsα{τKν, rDα,Kβs “ ÿ pτ,νq‰pH,βq KνDpsβ{ν˚sτqsα{τ, where Hdenotes the empty partition. 3. Part II: Combinatorial applications 3.1. Application to the skew Littlewood–Richardson rule. In this section, we present our first application of Theorem 1.1: a new proof of the skew Littlewood–Richardson rule as conjectured by Assaf and the second author [1] and proved by Lam, Lauve and Sottile [7]. As in [7], our starting point is (1.2). In [7], first an “algebraic skew Littlewood–Richardson rule” is derived, involving sums of products of Littlewood–Richardson coefficients. Then, the combinatorial skew Littlewood–Richardson rule is obtained by interpreting these Littlewood–Richardson coefficients as the number of semistandard Young tableaux with given rectification. Our proof fits more closely with the statement of the skew Littlewood–Richardson rule: we avoid going through the algebraic skew Littlewood–Richardson rule and use the interpretation of the Littlewood–Richardson coefficients as the number of semistandard Young tableaux with certain constraints on their content and reverse reading words. Our proof appears in Subsections 3.3 and 3.4 and is largely combinatorial. OPERATORS ON SYMMETRIC FUNCTIONS 17 For a positive integer kand a partition γ, the classical Pieri rule [10] gives a simple and beautiful expression for the product spkqsγas a sum of Schur functions. A k-horizontal (respectively k-vertical)strip is a skew shape with kboxes that has at most one box in each column (respectively row). The Pieri rule states that spkqsγ“ÿ p γ spγ, where the sum is over all partitions p γsuch that p γ{γis a k-horizontal strip. In [1], Assaf and the second author generalized the Pieri rule to the setting of skew shapes as follows: (3.1) spkqsγ{β“ k ÿ i“0 p´1qiÿ p γ, q β sp γ{q β, where the sum is over all partitions p γand q βsuch that p γ{γis a pk´iqhorizontal strip, and β{q βis an i-vertical strip. We will use the skew Pieri rule with k“1 in Section 3.5. For the next level of generality, it is natural to ask for a similarly combinatorial expression for sαsβ{γfor any partition α. Equation (2.26) gives one expression, but it does not mimic (3.1) in the sense that it does not give the answer as a signed sum of skew Schur functions. Instead, the skew Littlewood–Richardson rule [7] gives an expression for the even more general product sα{δsβ{γas a signed sum of skew Schur functions. In this section, we will derive the skew Littlewood–Richardson rule from (1.2) in the following way. In Subsection 3.3, we will use a combinatorial approach to obtain from (1.2) the skew Littlewood–Richardson rule in the case when δis empty, and then we will use a linearity argument to derive the result for general δin Subsection 3.4. 3.2. The combinatorial skew Littlewood–Richardson rule. In order to state the skew Littlewood–Richardson rule, we first need some terminology. As usual, a sequence of positive integers ωis said to be a lattice permutation if any prefix of ωcontains at least as many appearances of ias i`1, for all iě1. For a partition δ, we say that ωis a δ-lattice permutation if the word obtained by prefixing ωwith δ1copies of 1 followed by δ2copies of 2, etc., is a lattice permutation. We draw our Young tableaux in French notation, implying that the entries of an SSYT weakly increase along the rows and strictly increase up the columns. An anti-semistandard Young tableau (ASSYT) T1of shape α{βis a filling of the boxes of α{βso that the entries strictly decrease along the rows and weakly decrease up the columns. Equivalently, T1is an ASSYT if the tableau pT1 1qrobtained by transposing T1and then rotating it 180˝is an 18 E. BRIAND, P.R.W. MCNAMARA, R. ORELLANA, AND M. ROSAS SSYT. The reverse reading word of an SSYT T2is defined as usual as the word obtained by reading right-to-left along the rows of T2, taking the rows from bottom to top. In contrast, the reverse reading word of an ASSYT T1 is the word obtained by reading up the columns of T1, taking the columns from right-to-left. Equivalently, we can take the usual reverse reading word of the SSYT pT1 1qr. Given a pair of tableaux pT1, T2q, where T1is an ASSYT and T2is an SSYT, we define the reverse reading word of the pair as the concatenation of the reading word of T1with that of T2. We will encounter such pairs as in (3.2) below, where the entries in the bottom left form an ASSYT, and the entries above or to the right of the outlined skew shape form an SSYT. (3.2) 2 1 3 35 2 4 1 4 4 5 3 5 6 The reverse reading word of pT1, T2qshown in (3.2) is 21335425441365, which is certainly not a lattice permutation but is a 5321-lattice permutation. We are now ready to state the skew Littlewood–Richardson rule. Theorem 3.1 ([1, Conjecture 6.1]; [7, Theorem 3.2 and Remark 3.3(ii)]). For skew shapes α{δand γ{β, (3.3) sα{δsγ{β“ÿ T1PASSYTpβ{q βq T2PSSYTppγ{γq p´1q|β{q β|spγ{q β, where the sum is over all ASSYT T1of shape β{q βfor some q βĎβ, and SSYT T2of shape pγ{γfor some pγĚγ, with the following properties: (a) the combined content of T1and T2is the component-wise difference α´δ, and (b) the reverse reading word of pT1, T2qis a δ-lattice permutation. For example, the ASSYT and SSYT pair of (3.2) contribute ´s9953{1to the product s755431{5321s7541{33. Note that, when βand δare empty, we recover the classical Littlewood–Richardson rule. Indeed, this case corresponds to counting SSYT whose reverse reading words are lattice permutations; we call such SSYT LR-fillings. 3.3. Recovering a special case of the combinatorial skew Littlewood–Richardson rule. Our first step to reproving Theorem 3.1 is to start with (1.2) and show that it implies Theorem 3.1 in the case when δ“ H, the empty partition. Instead of (1.2), we work with the equivalent OPERATORS ON SYMMETRIC FUNCTIONS 19 identity in (2.26): sαsγ{β“ÿ λ p´1q|λ|Dβ{λ1psα{λsγq. First, let us examine the product sα{λsγfrom the right–hand side, and expand it in terms of Schur functions. Note that only those sνwith νĚγwill appear in the Schur expansion with nonzero coefficient. Thus we can write sα{λsγ“ÿ pγĚγ apγspγ, for some coefficients apγ. We have apγ“@spγˇˇsα{λsγD“@sp γ{γˇˇsα{λD“@sp γ{γsλˇˇsαD. The product spγ{γsλis equal to the skew Schur function of the shape pp γ{γq‘λ. (The notation ‘means that pγ{γis positioned so that its bottom-right corner box is immediately northwest of the top-left corner box of λ.) Therefore, the coefficient ap γis equal to the number of LR-fillings of that skew shape that have content α. Any LR-filling of that shape must just fill the ith row of λwith the number i, for all i. Thus apγequals the number of SSYT of shape pγ{γwhose reverse reading word is a λ-lattice permutation and whose content is the component-wise difference α´λ. Hence (2.26) is equivalent to (3.4) sαsγ{β“ÿ λ p´1q|λ|Dβ{λ1ÿ T2 spγ, where the second sum is over all SSYT T2of shape p γ{γfor some p γĚγ, and content α´λ, whose reverse reading word is a λ-lattice permutation. Next, we examine the term sβ{λ1. The coefficient of sνin this term is exactly the Littlewood–Richardson coefficient cβ λ1ν, which is only nonzero if νĎβ. Thus we wish to determine the coefficient cβ λ1q βof sq βin sβ{λ1when q βĎβ, which equals the number of LR-fillings of β{q βof content λ1. We claim that such fillings Tare in a shape-preserving bijection with ASSYT whose reverse readings word is a lattice permutation and whose content is λ (as opposed to content λ1previously). Indeed the bijection ψis defined by mapping the ith appearance (in the reverse reading word of the SSYT T) of the number jto the number i, for all iand j. For example, 3 4 233 122 1111 ÝÑ 3 1 321 521 4321 . Then one can check that ψhas the following necessary properties: 20 E. BRIAND, P.R.W. MCNAMARA, R. ORELLANA, AND M. ROSAS ˝The inverse of ψis given by the ASSYT analogue of ψ: map the jth appearance (in the reverse reading word, now in the ASSYT sense) of the number ito the number j, for all iand j. ˝The image ψpTqof an LR-filling Tis indeed an ASSYT whose reverse reading word is a lattice permutation. ˝Such a ψpTqmaps to an LR-filling under the inverse map. ˝Both ψand its inverse transpose the content partition. Thus (3.4) is equivalent to sαsγ{β“ÿ λ p´1q|λ|ÿ T1 Dq βÿ T2 sp γ“ÿ λ p´1q|λ|ÿ T1,T2 spγ{q β, where the relevant sums are over all T1and T2such that ˝T1is an ASSYT of content λ, whose reverse reading word is a lattice permutation, and with shape β{q βfor some q βĎβ, and ˝T2is an SSYT of content α´λ, whose reverse reading word is a λ-lattice permutation, and with shape pγ{γfor some pγĚγ. Note that T1tells us that |λ|“|β{q β|, and we have arrived at Theorem 3.1 in the case when δ“ H. 3.4. Recovering the full combinatorial skew Littlewood–Richardson rule. Our second step is to use a linearity argument to derive Theorem 3.1 for general δ. For this, observe that the coefficient of p´1q|β{q β|spγ{q βon the right-hand side of (3.3) is the number of pairs of tableaux pT1, T2qwith T1an ASSYT and T2a SSYT, fulfilling conditions (a) and (b) in Theorem 3.1. But T1being an ASSYT is equivalent to pT1 1qrbeing an SSYT, and the reverse reading word of the ASSYT T1is defined so that pT1 1qrhas the same reverse reading word as an SSYT. Therefore, the coefficient of p´1q|β{q β|spγ{q βon the right-hand side of (3.3) equals the number of SSYT of shape pp γ{γq‘pβ1{q β1qrand content αwhose reverse reading word is a δ– lattice permutation. This is the number of SSYT of shape pp γ{γq‘pβ1{q β1qr‘δ and content αwhose reverse reading word is a lattice permutation. By the Littlewood–Richardson rule, this quantity equals the coefficient of sαin the Schur expansion of sppγ{γq‘pβ1{q β1qr‘δ. This skew Schur function being equal to the product spγ{γspβ1{q β1qrsδmeans that the coefficient of sαis Asp γ{γspβ1{q β1qrsδˇˇˇsαE, which is equal to Asp γ{γspβ1{q β1qrˇˇˇsα{δE. OPERATORS ON SYMMETRIC FUNCTIONS 21 Therefore, (3.3) is equivalent to (3.5) sα{δsγ{β“ÿ pγ, q β1 p´1q|β{q β|Asp γ{γspβ1{q β1qrˇˇˇsα{δEsp γ{q β, where the sums are over all partitions p γand q β1such that p γĚγand q βĎβ. The key observation is that (3.5) is linear in sα{δ. Since the Schur functions form a basis for the space of symmetric functions, Equation (3.5) is equivalent to f¨sγ{β“ÿ pγ, q β1 p´1q|β{q β|Aspγ{γspβ1{q β1qrˇˇˇfEspγ{q β, for any symmetric function f, and any partitions βand γ. Hence, we see that in order to prove (3.3), it suffices to check the above equation for f“sα, for all partitions α. This is what we have done in Subsection 3.3. 3.5. A combinatorial interpretation for the Kronecker product of a skew Schur function by spn´1,1q.As another application of the identities of Section 2.7, our goal for this section is to derive a combinatorial formula for Kronecker products involving skew Schur functions. Let αbe a partition of n, and let us speak of partitions and their Young diagrams interchangeably. A well-known case of Kλis when λ“ p1q. We pause to describe an identity that we will generalize in Section 3.5 using the normal ordering relations. A corner of αis a box of αwhose removal results in another partition, and we denote by #cornerspαqthe number of corners of α. Denote by α´the set of partitions that result from removing a corner of α. Similarly, α`will denote the set of those partitions βsuch that αPβ´. We use α¯to denote the set of partitions not equal to αthat can be obtained by removing a corner of αand then adding a box to the result. Equivalently, α¯is the set of partitions that can be obtained from αby first adding a box and then removing a different box. For example, p31qhas two corners, and p31q¯“ tp4q,p22q,p211qu. We finished Subsection 2.6 by relating the combinatorial identity (3.6) Kp1qsα“ p#cornerspαq ´ 1qsα`ÿ βPα¯ sβ to the decomposition of the operators Kfin terms of the operators Up1qand Dp1q. We aim to generalize (3.6) to skew Schur functions. This leads to our next use of the identities of Section 2. Corollary 2.12 implies the relation rDθ, Kp1qs “ Dsθ{p1qs1, which gives Kp1qDθpsαq “ DθKp1qpsαq ´ Dsθ{p1qs1psαq. 22 E. BRIAND, P.R.W. MCNAMARA, R. ORELLANA, AND M. ROSAS Applying (3.6) and the fact that sθ{p1qs1“#cornerspθqsθ`ÿ φPθ¯ sφ, we get sα{θ˚spn´|θ|´1,1q“ p#cornerspαq´#cornerspθq´1qsα{θ`ÿ βPα¯ sβ{θ´ÿ φPθ¯ sα{φ, Thus we have an algebraic proof of the following result. Theorem 3.2. Suppose α$nand θ$kwith θĎα. Then sα{θ˚spn´k´1,1q“ p#cornerspαq´#cornerspθq´1qsα{θ`ÿ βPα¯ sβ{θ´ÿ φPθ¯ sα{φ. This identity begs for a combinatorial proof. We offer a proof which is “two-thirds” combinatorial. The part which is non-combinatorial makes use of (2.19), which in turn is proved using Littlewood’s identity (2.23). Second proof of Theorem 3.2. The proof proceeds in three stages. In the short first stage, which is the non-combinatorial one, we apply (2.19) to express sα{θ˚spn´k´1,1qin a form (3.7) not involving any Kronecker products. Then, using the skew Pieri rule (3.1), we reduce the problem to showing an identity (3.8) that is effectively purely about SSYT. This identity is proved in the third stage using jeu de taquin (first defined in [14]; can also be found in [12, 15], for example). First, we need to determine the result of applying Dp1qto the skew Schur function sα{θ. We have @sβˇˇDp1qsα{θD“@sβsp1qˇˇsα{θD“ÿ δPθ` xsβsδ|sαy“Csβˇˇˇˇˇÿ δPθ` sα{δG, and so Dp1qsα{θ“řδPθ`sα{δ. Applying (2.19) to sα{θ, we immediately deduce (3.7) sα{θ˚spn´k´1,1q“sp1qÿ δPθ` sα{δ´sα{θ. We next wish to apply the skew Pieri rule to sp1qřδPθ`sα{δ, but it will prove worthwhile to perform a preliminary step. By definition, sα{δ“0 unless δĎα, so it suffices to sum over the δ’s with δPθ`and δĎα. We write δPθ`αif δsatisfies these two conditions. So we now apply the skew Pieri rule (3.1) to sα{θ˚spn´k´1,1q“sp1qÿ δPθ`α sα{δ´sα{θ OPERATORS ON SYMMETRIC FUNCTIONS 23 to obtain sα{θ˚spn´k´1,1q“ÿ γPα` δPθ`α sγ{δ´ÿ δPθ`α φPδ´ sα{φ´sα{θ. Let us examine the second sum. For any δin θ`α, we can choose φ“θ. We see that the other φ’s that arise will be exactly those elements of θ¯that are contained in α. Therefore, sα{θ˚spn´k´1,1q“ÿ γPα` δPθ`α sγ{δ´ |θ`α|sα{θ´ÿ φPθ¯ sα{φ´sα{θ. Thus, to prove Theorem 3.2, it remains to show that (3.8) ÿ γPα` δPθ`α sγ{δ´ |θ`α|sα{θ“ p#cornerspαq ´ #cornerspθqqsα{θ`ÿ βPα¯ sβ{θ. Our main tool for proving the above identity is jeu de taquin but, as with our application of the skew Pieri rule, it is worthwhile to rewrite (3.8) in a slightly different form. Observe that, for any partition α, we have #cornerspαq “ |α`| ´ 1. For θĎα, denote the set of elements of θ`that are not contained in αby θ`αc; we can check that θ`αccan be non-empty only if α{θhas some empty rows or columns. We can now rewrite (3.8) as (3.9) ÿ γPα` δPθ`α sγ{δ“ p|α`|´|θ`αc|qsα{θ`ÿ βPα¯ sβ{θ. For intuition, we call the positions of those boxes of the form λ{αfor some λPα`the outside corners of α. Then the term |α`|´|θ`αc|is the number of outside corners of α, excluding those that are also outside corners of θ. See Example 3.3 below for a fully worked out example of the remainder of the proof. To prove (3.9) using jeu de taquin (jdt), consider an SSYT Tthat contributes to the left-hand side, meaning that Thas shape γ{δ, where γPα` and δPθ`α. Notice that the unique box bof δ{θis not an element of T, and that the unique box cof γ{αis in T. Perform a jdt slide of Tinto b, and let T1denote the resulting SSYT. There are three possibilities that can arise: (a) T1“T, meaning that there is no way to fill bunder a jdt slide of T. (b) T1contains b, and the single vacated box under the jdt slide is not c. (c) T1contains b, and the single vacated box under the jdt slide is c. By definition of jdt, Case (a) can happen if and only if bis a corner of γ. Since bPδĎαĂγ, the box bmust be a corner of α. Therefore, since bis the unique box of δ{θand is not an element of T1while cis the unique box of γ{αand is in T1, the shape γ{δof T1can be written in the equivalent form 24 E. BRIAND, P.R.W. MCNAMARA, R. ORELLANA, AND M. ROSAS β{θ, where βis obtained from αby removing band adding c. In particular, βPα¯, and so such T1s contribute part of the sum on the right of (3.9). We claim that Case (b) contributes the rest of the sum on the right of (3.9). We see that the shape of T1is obtained from the shape of Tby making exactly two changes: T1contains b, and a box c1different from chas been vacated. As a result, T1has shape β{θfor some βPα¯. To prove our claim from the start of the previous paragraph, let T1be an SSYT of shape β{θ, where βPα¯. We wish to show that T1arises as the image under jdt of exactly one Tfrom Cases (a) and (b). In short, the reason is that jdt slides are reversible, but let us be more precise. Suppose βis obtained from αby removing a box dand adding a different box e. Perform a jdt slide of T1into d. (Some references would call this a reverse jdt slide, since dis outside β.) There are two possibilities that can arise: (i) There is no way to fill dunder a jdt slide of T1. This can happen if and only if dis an outside corner of θ. Thus β{θcan be written in the equivalent form γ{δ, where γequals αwith eadded, and δ equals θwith dadded. Since dis a corner of α, it is also a corner of γ. These are exactly the conditions for T1to arise as an image under jdt of a Tfrom Case (a) (where in fact T“T1and our dhere corresponds to bin Case (a)). (ii) Performing a jdt slide of T1into dfills dand vacates an outside corner of θ. This is exactly the reverse of the jdt slide from Case (b) (where dis playing the role of c1). Thus T1arises as the image of a single Tfrom Cases (a) and (b). It remains to consider Case (c). Note that all T1in Case (c) are of shape α{θ. We would like to show that each T1is the image under jdt of kdistinct T, where k“ p|α`| ´ |θ`αc|q. This would show that the T1s from Case (c) together contribute the term p|α`|´|θ`αc|qsα{θfrom the right-hand side of (3.9), and (3.9) would be proved. So pick a T1from Case (c). Pick an outside corner cof α, and perform a (reverse) jdt slide of T1into c. If cis not an outside corner of θ, then this jdt slide will fill c, and the result will be an SSYT Tof shape γ{δwith γPα` and δPθ`α. These are exactly the conditions for T1to arise in Case (c). On the other hand, if cis an outside corner of αand also of θ, then T1 will remain fixed under the (reverse) jdt slide. Thus any Tthat maps to such a T1under a jdt slide must also have shape α{θ. Such a Tfrom the left-hand side of (3.9) does not exist, since Twould contain the single box γ{αwhereas T1does not. We conclude that each T1in Case (c) is the image of exactly kdistinct Ts, as required.  OPERATORS ON SYMMETRIC FUNCTIONS 25 Case (a) Case (b) Case (c) γ{δbcb cb c c1b c c1bc b c T bAB C b AB C b A B Cb ABCb A B C b A B C T1 dAB C d AB C C A B dC ABd B A Cc A c B C Table 3.1. The full set of skew shapes and SSYTs for Example 3.3. In the first row, the dashed boxes are those in δ, and the solid boxes are those in γ{δ. Lowercase letters correspond to notation in the proof of (3.9), while the uppercase A,B and Cdenote entries of the SSYTs that we assume satisfy the necessary inequalities but are otherwise any positive integers. Example 3.3. Suppose α“ p4,1,1qand θ“ p2,1q. We have α{θ“ The complete set of shapes γ{δ, and SSYT Tand T1from the proof of (3.9) are shown in Table 3.1. Deliberately omitted from the table is the scenario from the last paragraph of the proof, where cis an outside corner of both αand θ, which does not contribute to either side of (3.9). In this example, there is 1 “ |θ`αc|such situation, shown below. α{θ“c Acknowledgments. We thank Aaron Lauve for helpful comments. R. Orellana is grateful for the hospitality of the University of Sevilla and IMUS. E. Briand, P. McNamara and M. Rosas are grateful for the hospitality of the University of Rennes 1 and IRMAR in summer 2014. E. Briand and M. Rosas have been partially supported by the following projects: MTM2010–19336, MTM2013–40455–P, MTM2016–75024–P and