2023-07-20

## Cylindric Schur polynomials

The family of (skew) cylindric Schur functions were 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 Grassmanian. See also [McN06] for an introduction to this family of symmetric functions.

The cylindric Schur functions are a subset of skew affine Schur functions, which was proved by T. Lam, in [Prop. 33, Lam06a].

### Definition

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

A cylindric semi-standard 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 semi-standard 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.**

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 boxes under the shift action has been assigned an integer. Note that the second example shows that the skew Schur functions is 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 $d\geq 0.$ The cylindric shape is constructed as follows: Draw the outline of the partition $\lambda$ in the plane, but shifted $d$ steps right and $d$ steps down, then 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.

Every cylindric Schur polynomial can be obtained as a skew affine Schur function.

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

## 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 non-zero if and only of 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 conjecture 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 $\lambda, \mu,\nu subseteq \ell \times k,$ $|\lambda|+|\mu| = |\nu|+mn,$

\[ C_{\lambda,\mu}^{\nu,m} = \sum_{\rho} \sign(\rho/\nu) c^{\nu}_{\lambda,\mu}, \qquad \sign(\rho/\nu) \coloneqq \prod_{j} (-1)^{k-\mathrm{width}(r_j)}, \]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 fusion coefficients as seen in [Kor09]. For some positivity results on fusion coefficients, see [MS12].

Another relation notion are 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. NcNamara 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 recently answered in [Lee17] in the following theorem:

\begin{equation*} \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). \end{equation*}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, we have that $c^{\lambda/d/\mu}_{\nu/0/\emptyset} = C_{\mu,\nu}^{\lambda,d}.$

## Cylindric complete homogeneous polynomials

Introduced in [KP20].

## Cylindric elementary polynomials

Introduced in [KP20].

## See also

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

## References

- [BCF99] Aaron Bertram, Ionuţ Ciocan-Fontanine and William Fulton. Quantum multiplication of Schur polynomials. Journal of Algebra, 219(2):728–746, September 1999.
- [Kor09] Christian Korff. A combinatorial derivation of the Racah–Speiser algorithm for Gromov-Witten invariants. arXiv e-prints, 2009.
- [Kor20] Christian Korff. Cylindric hecke characters and Gromov–Witten invariants via the asymmetric six-vertex model. Communications in Mathematical Physics, 381(2):591–640, December 2020.
- [KP20] Christian Korff and David Palazzo. Cylindric symmetric functions and positivity. Algebraic Combinatorics, 3(1):191-247, 2020.
- [Lam06a] Thomas Lam. Affine Stanley symmetric functions. American Journal of Mathematics, 128(6):1553–1586, 2006.
- [Lee17] Seung Jin Lee. Positivity of cylindric skew Schur functions. arXiv e-prints, 2017.
- [McN06] Peter McNamara. Cylindric skew Schur functions. Advances in Mathematics, 205(1):275–312, September 2006.
- [MS12] Jennifer Morse and Anne Schilling. A combinatorial formula for fusion coefficients. 24th International Conference on Formal Power Series and Algebraic Combinatorics. Discrete Mathematics and Theoretical Computer Science, January 2012.
- [Pos05] Alexander Postnikov. Affine approach to quantum Schubert calculus. Duke Mathematical Journal, 128(3):473–509, June 2005.