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Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups

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Naoya Yamaguchi, Yuka Yamaguchi, Genki Shibukawa

Representation Theory · Submitted 28 Mar 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We also show that these special values naturally appear as the coefficients in the expansion of the $k$th power of the circulant determinant of order $n$ (the group determinant of the cyclic group of order $n$).

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arXiv ID2203.14422 (version 4)
Pith IDQRS2DCVQ, Pith's permanent record of this paper and its review
CitationGenki Shibukawa, Naoya Yamaguchi, Yuka Yamaguchi. "Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups." arXiv:2203.14422 (2022). https://pith.science/paper/QRS2DCVQ
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,995 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.59 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

In this paper, we study the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at \[ \zeta_{(n,k)} := ( 1, \zeta_n, \zeta_n^2, \dots, \zeta_n^{kn-1} ), \] where $\zeta_n$ is a primitive $n$th root of unity. We give explicit formulas for several classes of special values. We also show that these special values naturally appear as the coefficients in the expansion of the $k$th power of the circulant determinant of order $n$ (the group determinant of the cyclic group of order $n$). These results extend Ore's formulas for the case $k = 1$. Furthermore, we determine the number of terms in the $k$th power of the group permanent of the cyclic group of order $n$. This extends Brualdi and Newman's result for $k = 1$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper, we study the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at \[ \zeta_{(n,k)} := ( 1, \zeta_n, \zeta_n^2, \dots, \zeta_n^{kn-1} ), \] where $\zeta_n$ is a primitive $n$th root of unity. contribution 0.91 0.79 0.08 asserted 0.94
s1 We give explicit formulas for several classes of special values. result 0.67 0.96 0.16 asserted 1.00
s2 We also show that these special values naturally appear as the coefficients in the expansion of the $k$th power of the circulant determinant of order $n$ (the group determinant of the cyclic group of order $n$). result 0.92 0.95 0.44 asserted 1.00
s3 These results extend Ore's formulas for the case $k = 1$. interpretation 0.42 0.53 0.09 asserted 1.00
s4 Furthermore, we determine the number of terms in the $k$th power of the group permanent of the cyclic group of order $n$. result 0.83 0.96 0.13 asserted 1.00
s5 This extends Brualdi and Newman's result for $k = 1$. interpretation 0.35 0.57 0.06 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • T body sentences
  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In this paper, we study the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at \[ \zeta_{(n,k)} := ( 1, \zeta_n, \zeta_n^2, \dots, \zeta_n^{kn-1} ), \] where $\zeta_n$ is a primitive $n$th root of unity. We give explicit formulas for several classes of special values. We also show that these special values naturally appear as the coefficients in the expansion of the $k$th power of the circulant determinant of order $n$ (the group determinant of the cyclic group of order $n$). These results extend Ore's formulas for the case $k = 1$. Furthermore, we determine the number of terms in the $k$th power of the group permanent of the cyclic group of order $n$. This extends Brualdi and Newman's result for $k = 1$.

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3 arXiv papers cite this in 6 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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