Dominance of the Permanent among Immanants: A Study of Lieb's Conjecture and Related Inequalities
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Authors
Dai, Hongyang
Issue Date
2025
Type
Thesis
Language
en_US
Keywords
Alternative Title
Abstract
This thesis investigates a conjecture attributed to Lieb asserting that the permanent dominatesall normalized immanants over the cone of positive semidefinite Hermitian matrices. Beginning
with the definition of immanants via irreducible characters of the symmetric group Sn, the work
reviews and synthesizes key results that support the conjecture in special cases. Building upon
the foundational work of G.D. James and M.W. Liebeck, and P. Heyfron, we examine character-
based inequalities involving the permanent and determinant, and present a detailed exposition
of class functions constructed using the Lieb inequality.
Utilizing techniques from the representation theory of symmetric groups, including the induction
and restriction of characters, the Littlewood–Richardson Rule, and the construction of permu-
tation characters ξij , we demonstrate how these functions can be used to establish inequalities
between various immanants. Special attention is given to the structure of partitions with two
parts and hook-shaped partitions, culminating in a verification of Lieb’s Permanent Dominance
Conjecture in those cases.
This thesis also proposes a rigorous framework for representing the irreducible characters of the
symmetric group Sn by means of non-negative class functions. It highlights the limitations of the
methods developed by James and Liebeck, and Heyfron when applied to the immanent-ordering
problem for two-part partitions. By addressing these shortcomings, the study establishes several
new results, most notably Theorems 4.8 and 4.9.
Overall, the thesis advances the understanding of immanant inequalities and demonstrates the
power of representation-theoretic approaches in matrix analysis.