2020-09-09

## Loop Schur polynomials

The loop Schur functions were introduced in [Lam10LP12]. To define them, we first need to define loop symmetric functions.

### Loop symmetric functions

The following definitions are from [LP13]. We fix $n$ and $m$ and consider the array of variables

\[ x_i^{(r)} \quad 1\leq i \leq m \qquad r \in \setZ/n\setZ. \]The loop elementary symmetric functions and the loop homogeneous symmetric functions are defined as

\[ \elementaryE_{k}^{(r)}(\xvec) \coloneqq \sum_{1\leq i_1 \lt i_2 \lt \dotsc \lt i_k \leq m} x_{i_1}^{(r)} x_{i_2}^{(r+1)} \dotsb x_{i_r}^{(r+k-1)} \] \[ \completeH_{k}^{(r)}(\xvec) \coloneqq \sum_{1\leq i_1 \leq i_2 \leq \dotsc \leq i_k \leq m} x_{i_1}^{(r)} x_{i_2}^{(r-1)} \dotsb x_{i_r}^{(r-k+1)} \]where the upper indices is taken modulo $n.$ The superscript is commonly referred to as the color. Note that for $n=1,$ we recover the usual symmetric functions.

The loop elementary symmetric functions generate a ring — this is the ring of loop symmetric functions, $LSym_n.$

*Remark:* There are two versions defined in [LP12],
"whirl" and "curl". This is the "whirl" notation.
The difference is in how the color indices progress.

### Definition

Given a square $(i,j),$ (row, column) in a (skew) Young diagram, define its content as $c(i,j)\coloneqq i-j$ mod $n.$ If $T \in \SSYT(\lambda/\mu),$ let

\[ wr_r(T) \coloneqq \prod_{(i,j)\in \lambda/\mu} x_{T(i,j)}^{(r+c(i,j))}. \]The loop Schur polynomial is then defined as

\[ \schurS^{(r)}_{\lambda/\mu}(x_1,\dotsc,x_m) \coloneqq \sum_{T \in \SSYT(\lambda/\mu)} wt_r(T). \]**Example.**

For $\lambda=(3,2),$ $n=3$ and $m=2,$ we have that

\[ \schurS^{(2)}_{(3,2)}(x_1,x_2) = x_1^{(2)}x_1^{(1)}x_1^{(3)}x_2^{(3)}x_2^{(2)} + x_1^{(2)}x_1^{(1)}x_2^{(3)}x_2^{(3)}x_2^{(2)}. \]The two tableaux are

$1^{(2)}$ | $1^{(1)}$ | $1^{(3)}$ |

$2^{(3)}$ | $2^{(2)}$ |

$1^{(2)}$ | $1^{(1)}$ | $2^{(3)}$ |

$2^{(3)}$ | $2^{(2)}$ |

where the superscript indicate the color.

**Problem.**

The loop Schur functions do not span the ring of loop symmetric functions. Also, they are not linearly independent! It is an open problem, see [Lam10], to find an appropriate Schur-like basis (or a monomial-like basis) for $LSym_n.$

### Alternant quotient identity

As with the Schur polynomials, we can express loop Schur functions as a ratio of alternants. Define the following $m\times m$-matrix:

\[ (A_\alpha^{(r)})_{ij} \coloneqq t_{j,m} (x_m^{(r+m-1)} x_m^{(r+m-2)} \dotsb x_m^{(r+m-\alpha_i)}) \]where $t_{j,m}$ is a certain transposition under a birational $S_m$-action on $\setQ( x_i^{(j)}),$ see [Fri20] for details.

Then

\[ \schurS^{(r)}_{\lambda} = \frac{\left| A_{\lambda+\delta}^{(r)} \right|}{ \left| A_{\delta}^{(r)} \right| } \]where $\delta = (m-1,m-2,\dotsc,1,0).$

This also appears in [Lam10], but as pointed out in [Fri20], there is a typo in Lam's notes. The idea of the proof is attributed to Greg Anderson.

### Jacobi–Trudi identity

In [LP12], it is proved that

\[ \schurS^{(r)}_{\lambda/\mu} = \det\left( \completeH_{\lambda_i - \mu_j - i+j}^{(r-\mu_j+j-1)} \right) = \det\left( \elementaryE_{\lambda'_i - \mu'_j - i+j}^{(r-\mu'_j-j+1)} \right). \]These identities generalize the classical Jacobi-Trudi identities for Schur polynomials.

### Murnaghan–Nakayama rule

A Murnaghan–Nakayama rule was proved in [Ros13]. An alternative proof can be found in [Fri20].

Define the loop power-sum symmetric functions as

\[ \powerSum_k(x_1,\dotsc,x_m) \coloneqq \sum_{i=1}^m \left( x_i^{(1)}x_i^{(2)}\dotsb x_i^{(n)} \right)^k. \]**Theorem (See [Ros13]).**

Let $\lambda$ be a partition with at most $m$ parts and $k\geq 1.$ Then

\[ \powerSum_k(x_1,\dotsc,x_m) \schurS^{(r)}_{\lambda}(x_1,\dotsc,x_m) = \sum (-1)^{\mathrm{ht}(\mu/\lambda)} \schurS^{(r)}_{\lambda}(x_1,\dotsc,x_m) \]where the sum is over all partitions $\mu$ with at most $n$ parts, such that $\mu/\lambda$ is a ribbon of size $kn.$

For notation and terminology, see the Murnaghan–Nakayama rule for Schur functions.

## References

- [Fri20] Gabriel Frieden. A ratio of alternants formula for loop Schur functions. Journal of Combinatorics, 11(2):359–376, 2020.
- [Lam10] Thomas Lam. Loop symmetric functions and factorizing matrix polynomials. arXiv e-prints, 2010.
- [LP12] Thomas Lam and Pavlo Pylyavskyy. Total positivity in loop groups, I: Whirls and curls. Advances in Mathematics, 230(3):1222–1271, June 2012.
- [LP13] Thomas Lam and Pavlo Pylyavskyy. Intrinsic energy is a loop Schur function. Journal of Combinatorics, 4(4):387–401, 2013.
- [Ros13] Dustin Ross. The loop Murnaghan–Nakayama rule. Journal of Algebraic Combinatorics, 39(1):3–15, March 2013.