The peak algebra in noncommuting variables
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915518514987008 |
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| author | Aliniaeifard, Farid Li, Shu Xiao |
| author_facet | Aliniaeifard, Farid Li, Shu Xiao |
| contents | The well-known descent-to-peak map $Θ_{\mathrm{QSym}}$ for the Hopf algebra of quasisymmetric functions, $\mathrm{QSym}$, and the peak algebra $Π$ were originally defined by Stembridge in 1997. We introduce their noncommutative analogues, the labelled descent-to-peak map $Θ_{\mathrm{NCQSym}}$ for the Hopf algebra of quasisymmetric functions in noncommuting variables, $\mathrm{NCQSym}$, and the peak algebra in noncommuting variables $\mathrm{NC}Π$. Then, we define the Hopf algebra of Schur $Q$-functions in noncommuting variables. We show that our generalizations possess many properties analogous to their classical counterparts. Furthermore, we show that the coefficients in the expansion of certain elements of $\mathrm{NC}Π$ in the monomial basis of $\mathrm{NCQSym}$ satisfy the generalized Dehn-Sommerville equation of Bayer and Billera. In the end, we give representation-theoretic interpretations of the descent-to-peak map for the Hopf algebras of symmetric functions and noncommutative symmetric functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_12868 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The peak algebra in noncommuting variables Aliniaeifard, Farid Li, Shu Xiao Combinatorics 05E05, 05E10, 16T30, 20C30 The well-known descent-to-peak map $Θ_{\mathrm{QSym}}$ for the Hopf algebra of quasisymmetric functions, $\mathrm{QSym}$, and the peak algebra $Π$ were originally defined by Stembridge in 1997. We introduce their noncommutative analogues, the labelled descent-to-peak map $Θ_{\mathrm{NCQSym}}$ for the Hopf algebra of quasisymmetric functions in noncommuting variables, $\mathrm{NCQSym}$, and the peak algebra in noncommuting variables $\mathrm{NC}Π$. Then, we define the Hopf algebra of Schur $Q$-functions in noncommuting variables. We show that our generalizations possess many properties analogous to their classical counterparts. Furthermore, we show that the coefficients in the expansion of certain elements of $\mathrm{NC}Π$ in the monomial basis of $\mathrm{NCQSym}$ satisfy the generalized Dehn-Sommerville equation of Bayer and Billera. In the end, we give representation-theoretic interpretations of the descent-to-peak map for the Hopf algebras of symmetric functions and noncommutative symmetric functions. |
| title | The peak algebra in noncommuting variables |
| topic | Combinatorics 05E05, 05E10, 16T30, 20C30 |
| url | https://arxiv.org/abs/2506.12868 |