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#Vector-space properties of symmetric functions

The ring of symmetric functions is also a graded vector space with several natural extra structures. This page collects the Fock-space viewpoint, the Hall inner product, standard involutions and adjoint operators, and the Hopf algebra structure on \(\spaceSym.\)

#Fock space

In the context of symmetric functions, Fock space \(\mathcal{F}\) is a Hilbert space isomorphic to \(\spaceSym.\) It provides a framework where bases of symmetric functions are viewed as states created by operators acting on a vacuum. This point of view is particularly powerful for proving identities involving Schur functions via the Boson–Fermion correspondence, and for connecting symmetric functions to integrable systems. For background, see [Chap. 14, Kac90] and [Chap. 4, T00].

R. Fesler, M. A. Hahn, M. Karev, and H. Markwig introduce a two-parameter CJT-refinement of Jucys–Murphy theory acting on Fock space [FHKM25]. Their framework unifies Schur and zonal actions, conjecturally includes Jack actions, and gives cut-and-join recursions for \(b\)-Hurwitz numbers.

The space can be described in two equivalent ways:

The Bosonic picture: This picture identifies \(\mathcal{F}\) with the polynomial ring generated by the power-sum symmetric functions, \(\setC[\powerSum_1, \powerSum_2, \dotsc].\) The Heisenberg algebra acts on this space via operators \(\alpha_n\) for \(n \in \setZ \setminus \{0\}.\) For \(n \in \setZ \setminus \{0\},\) we define: \[\alpha_{-n} \cdot f = \powerSum_n \cdot f \quad \text{(creation for $n \gt{} 0$)}, \qquad \alpha_{n} \cdot f = n \frac{\partial f}{\partial \powerSum_n} \quad \text{(annihilation for $n \gt{} 0$)}.\] These satisfy the commutation relation \([\alpha_n, \alpha_m] = n \delta_{n+m,0} \mathrm{Id}.\) The vacuum state \(|0\rangle\) corresponds to the unit \(1 \in \spaceSym.\) In this picture, the Hall inner product arises naturally, making the adjoint of multiplication by \(\powerSum_n\) equal to the differential operator \(n \frac{\partial}{\partial \powerSum_n}.\)

The Fermionic picture: Here, \(\mathcal{F}\) is constructed as a semi-infinite wedge space \(\Lambda^{\frac{\infty}{2}} V,\) where \(V = \bigoplus_{i \in \setZ} \setC v_i.\) A basis is given by semi-infinite wedges of the form \[|S\rangle = v_{i_1} \wedge v_{i_2} \wedge v_{i_3} \wedge \dotsb\] where \(i_1 \gt{} i_2 \gt{} \dotsb\) is a decreasing sequence of integers such that \(i_k = -k+1\) for \(k\) large enough. This sequence corresponds to a partition \(\lambda\) via the relation \(\lambda_k = i_k - (-k+1).\) Combinatorially, the set \(\{i_1, i_2, \dotsc \}\) is often visualized as a Maya diagram.

The Boson–Fermion correspondence: There is a canonical isomorphism \(\sigma: \mathcal{F}_{\text{Fermion}} \to \mathcal{F}_{\text{Boson}}.\) Under this map, the basis wedge corresponding to a partition \(\lambda\) maps exactly to the Schur function \(\schurS_\lambda:\) \[\sigma(v_{i_1} \wedge v_{i_2} \wedge \dotsb ) = \schurS_\lambda(\xvec).\] This correspondence explains why Schur functions form an orthonormal basis: they correspond to the orthonormal basis vectors of the exterior algebra. Furthermore, the Bernstein creation operators can be viewed as the image of Fermionic creation operators acting on the Bosonic space.

#Generating families

V. S gives criteria for when a homogeneous sequence \(u_n\in\Lambda_n\) is an algebraically independent set of generators for the algebra of symmetric functions [S24]. The results apply to many standard families indexed by partitions or skew partitions, including monomial, forgotten, skew complete, skew elementary, Schur, skew Schur, Hall–Littlewood, \(q\)-Whittaker, Macdonald, and integral-form Macdonald functions.

#Hall inner product

Let \(\langle \cdot, \cdot \rangle\) be the inner product on symmetric functions such that \(\langle \powerSum_\lambda, \powerSum_\mu \rangle = \delta_{\lambda\mu}z_\lambda,\) that is, the power-sum functions form an orthogonal basis. Then \(\langle \schurS_\lambda, \schurS_\mu \rangle = \delta_{\lambda\mu},\) so that the Schur polynomials form an orthonormal basis for \(\spaceSym.\) See the preliminaries for the definition of \(z_\lambda.\)

Example

For \(\lambda=(2,1),\) \[\langle \powerSum_{21}, \powerSum_{21} \rangle = z_{21} = 2, \qquad \langle \schurS_{21}, \schurS_{21} \rangle = 1.\] This illustrates the different normalizations of the power-sum and Schur bases.

The following properties hold for the Hall inner product:

  • The monomial basis and the complete homogeneous basis are dual: \(\langle \monomial_\lambda, \completeH_\mu \rangle = \delta_{\lambda\mu}.\)

  • Any symmetric function \(f\) defines a map \(f:\spaceSym \to \spaceSym\) by multiplication. This gives an adjoint map \(f^{\perp}: \spaceSym \to \spaceSym\) called skewing by \(f\). That is, \[\langle f\cdot g, h \rangle = \langle g, f^{\perp} h \rangle.\]

    For example, multiplication by \(\schurS_\mu\) is adjoint to the linear map that sends \(\schurS_\lambda\) to \(\schurS_{\lambda/\mu}.\) In other words, \[\langle \schurS_\mu \schurS_\nu, \schurS_{\lambda} \rangle = \langle \schurS_\nu, \schurS_{\lambda/\mu} \rangle.\] We also have \(\powerSum_n^{\perp} = n \frac{\partial}{\partial \powerSum_n}.\)

  • The \(\omega\) involution is self-adjoint: \[\langle \omega(f), g \rangle = \langle f, \omega(g) \rangle\]

  • Multiplication and comultiplication are adjoint; see the Hopf algebra structure on \(\spaceSym.\)

  • Schur positivity for an operator \(T\) implies Schur positivity for the adjoint. More precisely, \[T(\schurS_\lambda) = \sum_{\mu} c_{\lambda \mu} \schurS_\mu \iff T^{\perp}(\schurS_\mu) = \sum_{\lambda} c_{\lambda \mu} \schurS_\lambda.\] This holds since the Schur functions constitute an orthonormal basis.

See also these other maps on symmetric functions.

#Involution on symmetric functions

There is a standard involution \(\omega\) on symmetric functions, defined by \(\omega(\elementaryE_\lambda) = \completeH_\lambda\) for all \(\lambda.\) It acts on the power-sum symmetric functions and the Schur polynomials as \[\omega(\powerSum_\lambda) = (-1)^{|\lambda|-\length(\lambda)}\powerSum_\lambda \quad \text{and} \quad \omega(\schurS_\lambda) = \schurS_{\lambda'}.\]

This involution has natural extensions to quasisymmetric functions by defining it on the Gessel quasisymmetric functions.

Lemma (From [BN20]).

Let \(f(t)\) be a formal power series (or polynomial) with \(f(0)=1.\) Then \[\omega \left( f(x_1)f(x_2) \dotsm \right) = \frac{1}{f(-x_1)f(-x_2) \dotsm}.\]

#Other maps on symmetric functions

Definition (The Adams operator and the Verschiebung operator).

The \(k\)th Adams operator on \(\spaceSym\) is the ring homomorphism \[f(\xvec) \mapsto f[ \powerSum_k ] = f(x_1^k, x_2^k, \dotsc ),\] which is a plethysm.

The adjoint operator to the Adams operator is the Verschiebung operator \(\phi_k,\) which is defined on the power-sum symmetric functions via \[\phi_k \powerSum_m(\xvec) = \begin{cases} k\cdot \powerSum_{m/k}(\xvec) & \text{ if } k \mid m \\ 0 & \text{ otherwise,} \end{cases}\] and then extended as a ring homomorphism.

Definition (The Bernstein operator).

The Bernstein operator \(\mathbf{B}_r\) is defined as \[\mathbf{B}_r \coloneqq \sum_{j \geq 0} (-1)^j \completeH_{r+j} \elementaryE^{\perp}_j.\] It is also called the creation operator because it satisfies \[\schurS_{\lambda} = \mathbf{B}_{\lambda_1} \mathbf{B}_{\lambda_{2}} \dotsb \mathbf{B}_{\lambda_{\ell}}(1).\] That is, we can create the Schur function \(\schurS_{\lambda}\) from \(1,\) and \(\mathbf{B}_r\) is the operator that adds a new top row of length \(r.\)

Definition (The hook filter).

We define the linear map \(\phi\) on the ring of quasisymmetric functions by its action on the Gessel fundamental basis. For \(S \subseteq [n],\) \[\phi(\gessel_{n+1,S}(\xvec)) = \begin{cases} t(t-1)^i & \text{if } S = \{i+1, i+2, \dotsc, n\}, \\ 0 & \text{otherwise.} \end{cases}\]

Proposition

For a partition \(\lambda \vdash (n+1),\) the map \(\phi\) acts as an exact filter for hook shapes on Schur functions: \[\phi(\schurS_\lambda(\xvec)) = \begin{cases} t(t-1)^i & \text{if } \lambda = (i+1, 1^{n-i}), \\ 0 & \text{otherwise.} \end{cases}\]

This map has an important role for the chromatic symmetric functions [Sta95]. In particular, it has the property that \(\phi(\elementaryE_\mu) = t^{\length(\mu)}.\)

#Hopf algebra

The algebra of symmetric functions admits a Hopf algebra structure, where the coproduct is defined as \[\Delta( \elementaryE_k ) = \sum_{j=0}^k \elementaryE_j \otimes \elementaryE_{k-j}\] For introductions to Hopf algebras and symmetric functions, see [Che14, GR14].

The Hopf algebra antipode on symmetric functions has the property that \[S( \schurS_{\lambda/\mu} ) = (-1)^{|\lambda/\mu|} \schurS_{\lambda'/\mu'}.\] For a combinatorial proof of this, see [CHL26].

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