Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions
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| Format: | Preprint |
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2025
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| _version_ | 1866911224006967296 |
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| author | Khare, Apoorva Matherne, Jacob P. Dizier, Avery St. |
| author_facet | Khare, Apoorva Matherne, Jacob P. Dizier, Avery St. |
| contents | In 2022, Huh-Matherne-Mészáros-St. Dizier showed that normalized Schur polynomials are Lorentzian, thereby yielding their continuous (resp. discrete) log-concavity on the positive orthant (resp. on their support, in type $A$ root directions). A reinterpretation of this result is that the characters of finite-dimensional simple representations of $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian (DL). In the same paper, these authors also showed that shifted characters of Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are DL.
In this work we extend these results to a larger family of modules that subsumes both of the above: we show that shifted characters of all parabolic Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. The proof involves certain graphs on $[n+1]$; more strongly, we explain why the character (i.e., generating function) of the Kostant partition function of any loopless multigraph on $[n+1]$ is Lorentzian after shifting and normalizing.
We then show that parabolic Vermas form a "maximal" class with log-concave (hence DL) characters. Namely, log-concavity fails in greater generality along three natural directions: (1) it does not hold for every simple Lie type, (2) nor for a larger universal family of highest weight modules, the higher order Verma modules, even in type $A$, and (3) it does not always hold for important generalizations of Schur polynomials: the Jack and Macdonald polynomials.
Finally, we extend these results to parabolic (i.e. "first order") and higher order Verma modules over the semisimple Lie algebras $\oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$. We also partially resolve a conjecture of Huh et al on the DL property for integral highest weight simple modules. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01623 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions Khare, Apoorva Matherne, Jacob P. Dizier, Avery St. Representation Theory Combinatorics Primary 05E10, 17B10, Secondary 33D52, 52B11 In 2022, Huh-Matherne-Mészáros-St. Dizier showed that normalized Schur polynomials are Lorentzian, thereby yielding their continuous (resp. discrete) log-concavity on the positive orthant (resp. on their support, in type $A$ root directions). A reinterpretation of this result is that the characters of finite-dimensional simple representations of $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian (DL). In the same paper, these authors also showed that shifted characters of Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are DL. In this work we extend these results to a larger family of modules that subsumes both of the above: we show that shifted characters of all parabolic Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. The proof involves certain graphs on $[n+1]$; more strongly, we explain why the character (i.e., generating function) of the Kostant partition function of any loopless multigraph on $[n+1]$ is Lorentzian after shifting and normalizing. We then show that parabolic Vermas form a "maximal" class with log-concave (hence DL) characters. Namely, log-concavity fails in greater generality along three natural directions: (1) it does not hold for every simple Lie type, (2) nor for a larger universal family of highest weight modules, the higher order Verma modules, even in type $A$, and (3) it does not always hold for important generalizations of Schur polynomials: the Jack and Macdonald polynomials. Finally, we extend these results to parabolic (i.e. "first order") and higher order Verma modules over the semisimple Lie algebras $\oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$. We also partially resolve a conjecture of Huh et al on the DL property for integral highest weight simple modules. |
| title | Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions |
| topic | Representation Theory Combinatorics Primary 05E10, 17B10, Secondary 33D52, 52B11 |
| url | https://arxiv.org/abs/2504.01623 |