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#Loop Schur polynomials

The loop Schur functions were introduced in [Lam10, LP12]. 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 \[\elementaryE_{k}^{(r)}(\xvec) \coloneqq \sum_{1\leq i_1 \lt i_2 \lt \dotsc \lt i_k \leq m} x_{i_1}^{(r)} x_{i_2}^{(r+1)} \dotsb x_{i_k}^{(r+k-1)}\] \[\completeH_{k}^{(r)}(\xvec) \coloneqq \sum_{1\leq i_1 \leq i_2 \leq \dotsc \leq i_k \leq m} x_{i_1}^{(r)} x_{i_2}^{(r-1)} \dotsb x_{i_k}^{(r-k+1)}\] where the upper indices are 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, \(LSym_n.\)

S. Iwao, K. Motegi, and R. Ohkawa obtain tensor Schur polynomials and \(q\)-deformed loop elementary symmetric functions as three-dimensional partition functions of tetrahedral \(L\)-operators [IMO26]. At \(q=0\) their construction gives products of Schur polynomials and recovers the Schur shuffle formula; at generic \(q\) it produces explicit deformations of the loop elementary functions.

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 \[\mathrm{wt}_r(T) \coloneqq \prod_{(i,j)\in \lambda/\mu} x_{T(i,j)}^{(r+c(i,j))}.\] The loop Schur polynomial is then defined as \[\schurS^{(r)}_{\lambda/\mu}(x_1,\dotsc,x_m) \coloneqq \sum_{T \in \SSYT(\lambda/\mu)} \mathrm{wt}_r(T).\]

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 superscripts indicate the colors.

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

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

Bibliography

  1. [Fri20]Gabriel Frieden. A ratio of alternants formula for loop Schur functions. Journal of Combinatorics, 11(2):359–376, 2020.
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  doi = {10.4310/joc.2020.v11.n2.a7},
  url2 = {https://doi.org/10.4310/joc.2020.v11.n2.a7},
  year = {2020},
  publisher = {International Press of Boston},
  volume = {11},
  number = {2},
  pages = {359--376},
  author = {Gabriel Frieden},
  title = {A ratio of alternants formula for loop {S}chur functions},
  journal = {Journal of Combinatorics}
}
  2. [IMO26]Shinsuke Iwao, Kohei Motegi and Ryo Ohkawa. Tetrahedral ${L}$-operators, tensor Schur polynomials and $q$-deformed loop elementary symmetric functions. arXiv:2604.22141v1, 2026.
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    @article{IwaoMotegiOhkawa2026x,
  author = {Shinsuke Iwao and Kohei Motegi and Ryo Ohkawa},
  title = {Tetrahedral ${L}$-operators, tensor {S}chur polynomials and
    $q$-deformed loop elementary symmetric functions},
  year = {2026},
  eprint = {2604.22141v1},
  url = {https://arxiv.org/abs/2604.22141v1},
  journal = {arXiv e-prints}
}
  3. [Lam10]Thomas Lam. Loop symmetric functions and factorizing matrix polynomials. arXiv:1012.1262, 2010.
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    @article{Lam2010,
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Title = {Loop symmetric functions and factorizing matrix polynomials},
Year = {2010},
Eprint = {1012.1262},
  url = {https://arxiv.org/abs/1012.1262},
journal = {arXiv e-prints}
}
  4. [LP12]Thomas Lam and Pavlo Pylyavskyy. Total positivity in loop groups, I: Whirls and curls. Advances in Mathematics, 230(3):1222–1271, June 2012.
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}
  5. [LP13]Thomas Lam and Pavlo Pylyavskyy. Intrinsic energy is a loop Schur function. Journal of Combinatorics, 4(4):387–401, 2013.
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    @article{LamPylyavskyy2013,
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  volume = {4},
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  pages = {387--401},
  author = {Thomas Lam and Pavlo Pylyavskyy},
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}
  6. [Ros13]Dustin Ross. The loop Murnaghan–Nakayama rule. Journal of Algebraic Combinatorics, 39(1):3–15, March 2013.
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  author = {Dustin Ross},
  title = {The loop {M}urnaghan--{N}akayama rule},
  journal = {Journal of Algebraic Combinatorics}
}

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