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#Cylindric Schur polynomials

The family of (skew) cylindric Schur functions was introduced by A. Postnikov in [Pos05]. This is motivated by the fact that the coefficients in the Schur expansion of such cylindric Schur functions are given by 3-point Gromov–Witten invariants of the Grassmannian. See also [McN06] for an introduction to this family of symmetric functions. R. M. Adin, V. Reiner, and Y. Roichman study cyclic descents for tableaux, including the cylindric descent framework [ARR18].

The cylindric Schur functions are a subset of skew affine Schur functions, which was proved by T. Lam in [Prop. 33, Lam06].

#Definition

A cylindric shape \(\lambda\) is an infinite lattice path in \(\setZ^2,\) invariant under shifts by \((n-m,m),\) where \(m \in [1,n-1].\) Let \(C^{n,m}\) be the set of such paths. Suppose \(\lambda,\mu \in C^{n,m}\) and \(\mu\) always lies weakly to the left of \(\lambda.\) Then we say that \(\mu \subseteq \lambda\) and that \(\lambda / \mu\) is a cylindric skew shape.

A cylindric semistandard tableau of shape \(\lambda / \mu\) is a sequence \[\mu = \lambda^0 \subseteq \lambda^1 \subseteq \dotsb \subseteq \lambda^{\ell} = \lambda\] such that each shape \(\lambda^i/\lambda^{i-1}\) contains at most one box in each column. Furthermore, we can think of placing the value \(i\) in the boxes determined by \(\lambda^i/\lambda^{i-1}.\) The weight \((a_1,a_2,\dotsc,a_{\ell})\) of the tableau is given by letting \(a_i\) be the number of boxes in any \(n-m\) consecutive columns of \(\lambda^i/\lambda^{i-1}.\)

The cylindric Schur function \(\schurCylindric_{\lambda/\mu}(\xvec)\) is then defined as \[\schurCylindric_{\lambda/\mu}(\xvec) = \sum_{T \in \mathrm{CSSYT}(\lambda/\mu)} \xvec^T\] where the sum is over all cylindric semistandard tableaux of shape \(\lambda/\mu.\) One can show via a variant of the Bender–Knuth involution that all \(\schurCylindric_{\lambda/\mu}(\xvec)\) are symmetric functions.

Example (Two cylindric semistandard tableaux).

In the following examples, \(n=7,\) \(m=3.\) This implies invariance when shifting \(3\) steps up and \(4\) steps to the right. We show two cylindric semistandard tableaux, of two different shapes, where exactly one representative of each orbit of boxes under the shift action has been assigned an integer. Note that the second example shows that the skew Schur functions are a proper subset of cylindric Schur functions.

              $\times $ $ \times$           $\times$ $ \times $ $\times$             $\times$ $ \times $ $\times$         $ 1 $ $ 1 $ $ \times $ $ \times$       $ 1 $ $ 2 $ $ 2$             $ 2 $ $ 3 $ $ 4$           $ \times$ $ 3 $ $ 5$                           $ \times$             $ \times$ $ \times$         $\times$ $\times$ $ \times$ $ \times$       $ 1$             $ 1 $ $ 2$         $ 2 $ $ 2 $ $ 3 $ $ 4$        

The first tableau has weight \((3,3,2,1,1)\) and the second one \((2,3,1,1).\) The Schur expansion of the first shape gives \[\schurCylindric_{\lambda/\mu} = \schurS_{433} + \schurS_{4321} - \schurS_{33211} + \schurS_{2221111} - \schurS_{211111111}.\]

A. Postnikov uses the notation \(\lambda/d/\mu\) to describe a cylindric shape, where \(\lambda\) and \(\mu\) are partitions and \(d\geq 0\) [Pos05]. The cylindric shape is constructed as follows: Draw the outline of the partition \(\lambda\) in the plane, shifted \(d\) steps right and \(d\) steps down, then draw the outline of \(\mu\) without shift, and extend both outlines periodically.

In the example above, the first shape can be described as \((6,3,3)/1/(6,3,1)\) and the second shape is just \((4,4,4)/0/(3,2,0),\) so that when \(d=0\) the shape is a regular skew shape.

P. McNamara’s running example [p. 12, McN06] illustrates another way to read off this notation. For a cylindric shape on the cylinder with \(k=3\) and \(n-k=4,\) one first chooses a fundamental strip of representatives. In that strip the lower boundary determines \(\mu=(2,1),\) while the finite skew shape has outer partition \(\Lambda=(4,4,4,4,2,1,1).\) Removing the top \(7\)-ribbon twice gives \(\Lambda[-1]=(4,4,4,1)\) and then \(\Lambda[-2]=(3,3).\) Hence this cylindric shape is described as \((3,3)/2/(2,1).\)

For the poset structure on cylindric diagrams and its connection to root systems and the Bruhat order, see [NST23].

Every cylindric Schur polynomial can be obtained as a skew affine Schur function. This implies, in particular, that the support of the monomials in a skew Schur function forms an \(M\)-convex set.

Note that cylindric Schur polynomials are not in general special cases of \((P,w)\)-partitions; see [McN06]. This is easily seen from the fact that all \((P,w)\)-partitions are positive in the fundamental quasisymmetric basis, while most cylindric Schur functions are not.

I. Siegl defines cylindric \(P\)-tableaux for \((3+1)\)-free posets and corresponding \(P\)-analogues of cylindric Schur functions [Sie25]. The resulting determinantal functions are weight generating functions for cylindric \(P\)-tableaux and imply positivity for certain sums of elementary-expansion coefficients of chromatic symmetric functions of incomparability graphs.

#Affine bounded Littlewood identities

In [HKKO23], the authors study bounded Littlewood identities where the cylindric Schur functions appear. They give proofs of affine Littlewood identities , which are analogues of the Littlewood identities for cylindric Schur functions.

#Equivariant Quantum Pieri rule

For an equivariant quantum Pieri rule for the Grassmannian on cylindric shapes, see [BEMT22].

#Toric Schur polynomials

A shape \(\lambda/\mu \in C^{n,m}\) is called toric if every column contains at most \(m\) boxes. Cylindric Schur functions indexed by such shapes are referred to as toric Schur polynomials. All the usual skew shapes are toric.

The specialization \[\schurCylindric_{\lambda/d/\mu}(\xvec)(x_1,\dotsc,x_m) \coloneqq \schurCylindric_{\lambda/d/\mu}(x_1,\dotsc,x_m,0,\dotsc)\] is nonzero if and only if the shape \(\lambda/d/\mu\) is toric. These polynomials, in a finite set of variables, are referred to as toric Schur polynomials.

#Schur expansion

Postnikov proves that when \(\lambda/d/\mu\) is a toric shape in \(C^{n,m}\) the coefficients \(C_{\mu,\nu}^{\lambda,d}\) in the Schur expansion \[\schurCylindric_{\lambda/d/\mu}(x_1,\dotsc,x_m) = \sum_\nu C_{\mu,\nu}^{\lambda,d} \schurS_\nu(x_1,\dotsc,x_m)\] are non-negative integers given by certain Gromov–Witten invariants.

Problem (Postnikov, [Pos05]).

It is an open problem to find a (positive) combinatorial rule for \(C_{\mu,\nu}^{\lambda,d}.\) Note that such a rule would generalize the classical Littlewood–Richardson rule.

Postnikov also introduces a certain family of toric Specht modules and conjectures that they decompose into irreducible Specht modules with multiplicities given by the \(C_{\mu,\nu}^{\lambda,d}.\)

The coefficients \(C_{\mu,\nu}^{\lambda,d}\) can be computed as an alternating sum of Kostka coefficients, see [Cor. 1.4, Kor09].

Another formula uses an alternating sum of regular Littlewood–Richardson coefficients; see [Eq. (21), BCF99]. They show that for \(\lambda, \mu,\nu \subseteq \ell \times k,\) \(|\lambda|+|\mu| = |\nu|+mn,\) \[C_{\lambda,\mu}^{\nu,m} = \sum_{\rho} \sign(\rho/\nu) c^{\rho}_{\lambda,\mu}, \qquad \sign(\rho/\nu) \coloneqq \prod_{j} (-1)^{k-\mathrm{width}(r_j)},\] where the sum is over all diagrams \(\rho\) obtained from \(\nu\) by adding \(m\) ribbons, each ribbon \(r_j\) containing \(n\) boxes and starting from the first row.

The \(C_{\mu,\nu}^{\lambda,d}\) are closely related to fusion coefficients, as seen in [Kor09]. For some positivity results on fusion coefficients, see [MS12].

Another related notion is the quantum Kostka coefficients.

#Cylindric Schur positivity

P. McNamara [McN06] asked for which cylindric shapes \(\schurCylindric_{\lambda/d/\mu}(\xvec)\) are Schur-positive as proper symmetric functions in an infinite alphabet, and conjectured that each cylindric skew Schur function can be positively expanded in certain non-skew cylindric Schur functions, with coefficients being Gromov–Witten invariants. P. McNamara proves that only ordinary skew Schur functions are Schur-positive. In fact, these are the only cylindric Schur functions that are positive in the Gessel fundamental quasisymmetric basis; see [Thm. 5.7, McN06].

McNamara’s conjecture was answered in [Lee17] by the following theorem: \[\schurCylindric_{\lambda/d/\mu}(\xvec) = \sum_{\nu \in C^{n,m}, e\geq 0} c^{\lambda/d/\mu}_{\nu/e/\emptyset} \schurCylindric_{\nu/e/\emptyset}(\xvec).\] The coefficients are all non-negative and satisfy \(c^{\lambda/d/\mu}_{\nu/e/\emptyset} = c^{\lambda/d-1/\mu}_{\nu/e-1/\emptyset}\) whenever \(e \gt 0.\) Furthermore, \(c^{\lambda/d/\mu}_{\nu/0/\emptyset} = C_{\mu,\nu}^{\lambda,d}.\)

For a short proof, see [Dob26], where the structure coefficients are interpreted using fusion coefficients.

#Cylindric complete homogeneous polynomials

The cylindric complete homogeneous polynomials were introduced in [KP20].

#Cylindric elementary polynomials

The cylindric elementary polynomials were introduced in [KP20].

#See also

C. Korff discusses a connection with a 6-vertex model and Hecke characters, see [Kor20].

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