[Paper Review] Commutation and normal ordering for operators on symmetric functions

Emmanuel Briand, Peter R. W. McNamara+2 more|arXiv (Cornell University)|Sep 9, 2015

Advanced Combinatorial Mathematics13 references3 citations

TL;DR

This paper establishes commutation and normal ordering relations for four families of operators—multiplication, skewing, Kronecker product, and a novel $¯{K}$ operator—acting on symmetric functions. Using Schur-generating series and formal power series techniques, it derives explicit commutator identities and proves uniqueness of normal ordering for specific operator pairs, offering a new proof of the skew Littlewood–Richardson rule and a new identity for Kronecker products with skew Schur functions.

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.

Motivation & Objective

• To derive commutation relations between four families of operators on symmetric functions: $U_\lambda$ (multiplication), $D_\lambda$ (skewing), $K_\lambda$ (Kronecker product), and $\overline{K}_\lambda$ (a new Kronecker-type operator).
• To develop a uniform method using Schur-generating series to systematically analyze and prove operator identities.
• To establish normal ordering relations and prove their uniqueness for specific operator pairs, resolving ambiguities in non-commutative operator expressions.
• To apply the results to prove the skew Littlewood–Richardson rule and derive a new identity involving Kronecker products with skew Schur functions.

Proposed method

• Define four operator families: $U_f$ (multiplication by $f$), $D_f$ (adjoint of $U_f$ under Hall inner product), $K_f$ (Kronecker multiplication), and $\overline{K}_f$ (a generalized Kronecker operator defined via Jacobi–Trudi determinants).
• Introduce the Schur-generating series $\sum_\lambda s_\lambda[A] P_\lambda$ for each operator $P$, which encodes the action of $P$ on symmetric functions via symmetric function bases.
• Use formal power series operations and known identities in symmetric function theory to derive and prove the commutation relations between all pairs of operators.
• Prove that finite expansions in normal order are unique for pairs $(U,D)$, $(D,U)$, $(U,K)$, $(K,D)$, $(U,\overline{K})$, and $(\overline{K},D)$, while non-unique for others like $(K,U)$.
• Apply the formalism to re-derive the skew Littlewood–Richardson rule by analyzing the structure of $K_\beta U_\alpha$ and $U_\alpha K_\beta$ in terms of Schur functions.
• Use the theory of reverse jdt slides and jeu de taquin to prove a key identity involving skew Schur functions and Kronecker coefficients, confirming the new identity for $K_f(s_{\alpha/\beta})$.

Experimental results

Research questions

• RQ1 What are the commutation relations between the operators $U_\lambda$, $D_\lambda$, $K_\lambda$, and $\overline{K}_\lambda$ on the ring of symmetric functions?
• RQ2 Can any of these operators be expressed as linear combinations of others via normal ordering, and is such an expression unique?
• RQ3 How can the Schur-generating series formalism be used to derive and unify commutation identities across all operator pairs?
• RQ4 Can the new operator $\overline{K}_\lambda$ be used to give a new proof of the skew Littlewood–Richardson rule?
• RQ5 What is the structure of the Kronecker product $K_f(s_{\alpha/\beta})$, and can it be expressed in terms of other symmetric functions?

Key findings

• The paper establishes a complete set of six commutation identities for all pairs of the four operator families, including $D_\beta U_\alpha = \sum_\lambda U_{\alpha/\lambda} D_{\beta/\lambda}$ and $\overline{K}_\beta U_\alpha = \sum_{\tau,\nu} U_{(s_{\beta/\nu} * s_\tau) s_{\alpha/\tau}} \overline{K}_\nu$.
• Finite normal ordering expansions are unique for the pairs $(U,D)$, $(D,U)$, $(U,K)$, $(K,D)$, $(U,\overline{K})$, and $(\overline{K},D)$, but not for $(K,U)$ or $(D,K)$, as shown by counterexamples like $K_{p_2}U_{p_1} = 0 = K_2U_1 = K_{1,1}U_1$.
• The skew Littlewood–Richardson rule is reproven using the operator formalism, showing that the structure coefficients arise naturally from the commutation of $U_\alpha$ and $K_\beta$.
• A new identity is proven: $K_f(s_{\alpha/\beta}) = \sum_{\tau,\nu} s_{(s_{\beta/\nu} * s_\tau) s_{\alpha/\tau}} \overline{K}_\nu$, which expresses the Kronecker product of a symmetric function with a skew Schur function in terms of the $\overline{K}$ operators.
• The use of Schur-generating series provides a uniform and elegant method to derive all identities, with the operators recoverable via scalar products in the symmetric function ring.
• The theory of reverse jdt slides and jeu de taquin is used to prove a key identity involving the number of preimages under jdt, confirming the coefficient $|\alpha^+| - |\theta^{+\alpha^c}|$ in the expansion of $K_\beta U_\alpha$.