Pith. Sign in

RECORD

Ribbon Operators and Hall-Littlewood Symmetric Functions

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Mike Zabrocki

Quantum Algebra · Submitted 22 Aug 2000

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

The authors' main claim We show here that $H_{1^k}^q$ has truly remarkable properties, in particular that it is a Rodriguez operator that adds a column to the Hall-Littlewood symmetric functions.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv IDmath/0008163 (version 1)
Pith ID2FTEQMT5, Pith's permanent record of this paper and its review
CitationMike Zabrocki. "Ribbon Operators and Hall-Littlewood Symmetric Functions." arXiv:math/0008163 (2000). https://pith.science/paper/2FTEQMT5
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,290 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.32 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.60 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Given a partition $\la = (\la_1, \la_2, ... \la_k)$, let $\la^{rc} = (\la_2-1, \la_3-1, ... \la_k-1)$. It is easily seen that the diagram $\la\slash \la^{rc}$ is connected and has no $2 \times 2$ subdiagrams which we shall refer to as a ribbon. To each ribbon $R$, we associate a symmetric function operator $S^R$. We may define the major index of a ribbon $maj(R)$ to be the major index of any permutation that fits the ribbon. This paper is concerned with the operator $H_{1^k}^q = \sum_R q^{maj(R)} S^R$ where the sum is over all $2^{k-1}$ ribbons of size $k$. We show here that $H_{1^k}^q$ has truly remarkable properties, in particular that it is a Rodriguez operator that adds a column to the Hall-Littlewood symmetric functions. We believe that some of the tools we introduce here to prove our results should also be of independent interest and may be useful to establish further symmetric function identities.

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 Given a partition $\la = (\la_1, \la_2, ... background 0.98 0.15 0.01 unclear 0.88
s1 \la_k)$, let $\la^{rc} = (\la_2-1, \la_3-1, ... \la_k-1)$. background 0.93 0.42 0.01 asserted 0.90
s2 It is easily seen that the diagram $\la\slash \la^{rc}$ is connected and has no $2 \times 2$ subdiagrams which we shall refer to as a ribbon. background 1.00 0.37 0.02 asserted 1.00
s3 To each ribbon $R$, we associate a symmetric function operator $S^R$. contribution 0.49 0.74 0.03 asserted 1.00
s4 We may define the major index of a ribbon $maj(R)$ to be the major index of any permutation that fits the ribbon. background 0.83 0.37 0.02 asserted 0.93
s5 This paper is concerned with the operator $H_{1^k}^q = \sum_R q^{maj(R)} S^R$ where the sum is over all $2^{k-1}$ ribbons of size $k$. contribution 0.80 0.32 0.09 asserted 0.95
s6 We show here that $H_{1^k}^q$ has truly remarkable properties, in particular that it is a Rodriguez operator that adds a column to the Hall-Littlewood symmetric functions. result 0.99 0.95 0.90 asserted 1.00
s7 We believe that some of the tools we introduce here to prove our results should also be of independent interest and may be useful to establish further symmetric function identities. future work 0.96 0.20 0.03 hedged possibility 0.95

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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

Given a partition $\la = (\la_1, \la_2, ... \la_k)$, let $\la^{rc} = (\la_2-1, \la_3-1, ... \la_k-1)$. It is easily seen that the diagram $\la\slash \la^{rc}$ is connected and has no $2 \times 2$ subdiagrams which we shall refer to as a ribbon. To each ribbon $R$, we associate a symmetric function operator $S^R$. We may define the major index of a ribbon $maj(R)$ to be the major index of any permutation that fits the ribbon. This paper is concerned with the operator $H_{1^k}^q = \sum_R q^{maj(R)} S^R$ where the sum is over all $2^{k-1}$ ribbons of size $k$. We show here that $H_{1^k}^q$ has truly remarkable properties, in particular that it is a Rodriguez operator that adds a column to the Hall-Littlewood symmetric functions. We believe that some of the tools we introduce here to prove our results should also be of independent interest and may be useful to establish further symmetric function identities.

Human review

How Pith checks papers → · Full audit of this paper →

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.