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Real gamma distribution on analytic bundles of flag varieties

Haoming Wang

TL;DR

The paper develops a real gamma distribution framework on analytic bundles over flag varieties by introducing four nested matrix normal families $T_1$, $T_{1/2}$, $T_2$, and $T_3$ that encode tensor-structured precision matrices. It builds a comprehensive toolkit—including the normal map, hypergeometric functions of matrix arguments, and explicit density/MGF formulas—to analyze the distributions of quadratic forms, sample covariances, and eigenvalues on flag varieties. Key contributions include central and noncentral density forms for heterogeneous quadratic forms, MGFs in terms of ${}_1F_0$ and ${}_0F_1$, and the joint eigenvalue distributions with flag-theoretic interpretations, culminating in a classification of double and single flag structures. The work bridges multivariate gamma theory, hypergeometric matrix functions, and the geometry of flag bundles, with potential extensions to complex cases and broader spectral decompositions in high-dimensional settings.

Abstract

This paper introduces four matrix normal distributions extending the separable covariance $ \varPhi \otimes \varPsi$ with potentially variable-level ($ \varPsi$) and/or sample-level ($ \varPhi$) correlations. The joint distribution of sample variances and covariances, leading to the product-moment distribution, is considered when precision matrices admit a specific tensor form. Several well-known results, including the non-central Wishart distribution and normal quadratic forms, now appear as corollaries. Moreover, we propose a conjecture concerning the analytic continuation of the real gamma distribution. By applying these results to the flag varieties, we classify the double flag and the single flag.

Real gamma distribution on analytic bundles of flag varieties

TL;DR

The paper develops a real gamma distribution framework on analytic bundles over flag varieties by introducing four nested matrix normal families , , , and that encode tensor-structured precision matrices. It builds a comprehensive toolkit—including the normal map, hypergeometric functions of matrix arguments, and explicit density/MGF formulas—to analyze the distributions of quadratic forms, sample covariances, and eigenvalues on flag varieties. Key contributions include central and noncentral density forms for heterogeneous quadratic forms, MGFs in terms of and , and the joint eigenvalue distributions with flag-theoretic interpretations, culminating in a classification of double and single flag structures. The work bridges multivariate gamma theory, hypergeometric matrix functions, and the geometry of flag bundles, with potential extensions to complex cases and broader spectral decompositions in high-dimensional settings.

Abstract

This paper introduces four matrix normal distributions extending the separable covariance with potentially variable-level () and/or sample-level () correlations. The joint distribution of sample variances and covariances, leading to the product-moment distribution, is considered when precision matrices admit a specific tensor form. Several well-known results, including the non-central Wishart distribution and normal quadratic forms, now appear as corollaries. Moreover, we propose a conjecture concerning the analytic continuation of the real gamma distribution. By applying these results to the flag varieties, we classify the double flag and the single flag.
Paper Structure (9 sections, 19 theorems, 56 equations, 1 table)

This paper contains 9 sections, 19 theorems, 56 equations, 1 table.

Table of Contents

  1. Introduction
  2. Analytic bundle
  3. Hypergeometric function
  4. Normal map
  5. Quadratic Form
  6. The moment generating function
  7. The distribution of characteristic roots
  8. Examples for $k=3$
  9. Conclusion

Key Result

Proposition 2.1

${\rm End}_{S_k}(V^{\otimes k}) \cong \mathbb{C}[GL(V)]$ and ${\rm End}_{GL(V)}(V^{\otimes k}) \cong \mathbb{C}[S_k]$.

Theorems & Definitions (35)

  • Proposition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 4.1
  • ...and 25 more