Gelfand–Tsetlin Subalgebra in U_q(gl_n)

• The Gelfand–Tsetlin subalgebra is a maximal commutative subalgebra in classical and quantum enveloping algebras, encoding key combinatorial data for constructing representation bases.
• The explicit quantum generators, derived via a generating function and analyzed with the De Concini–Kac filtration, reveal unique leading terms that ensure algebraic independence.
• This structure underpins the classification of Gelfand–Tsetlin modules, linking classical combinatorial techniques to quantum group representation and Galois order theory.

The Gelfand–Tsetlin subalgebra is a maximal commutative subalgebra present in enveloping algebras of classical and quantum Lie algebras, such as $U(\mathfrak{gl}_n)$ or its quantized analogue $U_q(\mathfrak{gl}_n)$. It is generated by the centers of a canonical sequence of nested subalgebras, and its explicit generators encode combinatorial data central to the construction of Gelfand–Tsetlin bases and modules. In the quantum case, the analysis of the subalgebra's generators via the De Concini–Kac filtration yields critical insight into their algebraic independence and the graded structure of the enveloping algebra (Futorny et al., 2012).

1. Definition and Structure of the Gelfand–Tsetlin Subalgebra

The construction begins with a strictly nested chain of subalgebras

$$U(1) \subset U(2) \subset \cdots \subset U(n) = U_q(\mathfrak{gl}_n)$$

where each $U(r)$ can be identified with $U_q(\mathfrak{gl}_r)$; $U_q(\mathfrak{gl}_n)$ denotes the quantized enveloping algebra. For each $r$, the center $Z_r$ of $U(r)$ is generated by central elements known as quantum determinants, whose coefficients arise in explicitly constructed polynomials (see formula (2.10) in (Futorny et al., 2012)).

The Gelfand–Tsetlin subalgebra $\mathcal{I}_q$ is the commutative subalgebra generated by all these centers: $$\mathcal{I}_q = \langle Z_1, Z_2, ..., Z_n \rangle \subset U_q(\mathfrak{gl}_n).$$ This subalgebra generalizes the classical GT subalgebra, serving as the spectral data source for Gelfand–Tsetlin bases and modules.

2. Explicit Generators and the De Concini–Kac Filtration

In $U_q(\mathfrak{gl}_n)$, the classical notion of GT generators is replaced by quantum counterparts $d_{r,s}$, defined as the coefficients in a generating function: $$Z_r(u) = \sum_{s=0}^r (-1)^s d_{r,s} q^{o_{r,s}} u^{-s},$$ where $o_{r,s}$ is an explicitly defined exponent and $u$ is a formal parameter (see formulas (2.12) and (2.13) in (Futorny et al., 2012)). Each $d_{r,s}$ is given, up to sign, by an alternating sum over permutations in the symmetric group $S_r$, with terms indexed by combinatorial patterns resembling the classical Gelfand–Tsetlin arrays.

The De Concini–Kac filtration is a graded structure on $U_q(\mathfrak{gl}_n)$, derived from a fixed reduced decomposition of the longest Weyl group element (formula (2.1)), and assigning degrees to PBW monomials via weight and combinatorial data (see formulas (2.4)–(2.5)). This filtration has the property that for any $a, b$ in the algebra, the degree satisfies $d(ab) = d(a) + d(b)$. The associated graded algebra is a domain, which underpins proofs of algebraic independence.

3. Computation of Leading Terms with Respect to Filtration

A central result from (Futorny et al., 2012) is the explicit determination of the leading monomial of each generator $d_{r,s}$ in the filtration (Theorem 3.1). Although $d_{r,s}$ is a sum over permutations, the leading contribution uniquely comes from the permutation $\tau = (1\,2\,\ldots\,r)^{(s)}$ that simultaneously maximizes two aspects of the total degree: the height (sum over $i - \tau(i)$) and the degree contributions from root vector exponents.

The leading term takes the form: $$\operatorname{lt}(d_{r,s}) = c \cdot t_{1,\epsilon_1} t_{2,\epsilon_2} \cdots t_{r,\epsilon_r}$$ where $c\in\mathbb{C}^*$, $\epsilon_i$ encodes the choice between exceedance and anti-exceedance according to combinatorial criteria—specifically, exactly $s$ indices have $\tau(i) < i$.

More explicitly, an example formula (see formula (3.1)) is: $$\operatorname{lt}(d_{r,s}) = c \cdot t_{1,1} t_{2,3} \ldots t_{r,r-s} \cdot t_{r-s+1,r-s+1} \ldots t_{r,r}.$$ The permutation's height is $2s(r-s)$, uniquely characterizing the dominant term (Lemma 3.5).

4. Algebraic Independence and Galois Order Perspective

Demonstrating a unique leading monomial for each $d_{r,s}$ immediately establishes that the generators are algebraically independent over the base field (see Lemma 2.3). Consequently, the quantum Gelfand–Tsetlin subalgebra $\mathcal{I}_q$ is a polynomial algebra, modulo possible localizations. This property is instrumental in identifying $U_q(\mathfrak{gl}_n)$ as a Galois ring—a notion that supports developments in the quantum version of the Gelfand–Kirillov conjecture.

The presence of a Galois order structure connects the representation theory of quantum groups with deep results in noncommutative invariant theory, and enables the explicit construction and classification of quantum Gelfand–Tsetlin modules.

5. Applications in Module Theory and Representation Theory

The quantum Gelfand–Tsetlin subalgebra acts by scalars on "Gelfand–Tsetlin modules," whose bases are combinatorially parametrized. Understanding the filtration and leading term structure provides refined control over the spectral decomposition of such modules. The resultant modules have weight-space decompositions labeled by Gelfand–Tsetlin patterns, which correspond to spectral values of the elements of $\mathcal{I}_q$.

This structure is fundamental for the explicit calculation of spectra, the paper of branching rules, and the classification of irreducible representations. It also conceptualizes the transfer of combinatorial structures from the classical to the quantum setting, supporting further generalizations.

6. Filtration Methods, Graded Structures, and Classical–Quantum Correspondence

The De Concini–Kac filtration serves not only as a technical device for proofs but also as an approximation technique. The associated graded algebra linearizes the structure of $U_q(\mathfrak{gl}_n)$, making it amenable to commutative techniques and simplifying the analysis of modules and subalgebras. Such filtration methods are widely utilized in both commutative and noncommutative invariant theory.

The similarities in combinatorial pattern, filtration method, and spectral properties between the classical and quantum cases highlight profound connections. These parallels substantiate the use of classical representation-theoretic intuition within the quantum theory framework, with immediate applications in constructing quantum analogues of Gelfand–Tsetlin modules.

7. Summary of Essential Formulas

The main structural formulas encoding the Gelfand–Tsetlin subalgebra and its properties include:

• The central generating series:

$$Z_r(u) = \sum_{s=0}^r (-1)^s d_{r,s} q^{(\cdots)} u^{-s}$$

(formula (2.12))

• Explicit combinatorial expression for $d_{r,s}$:

$$d_{r,s} = \sum_{(k_1, ..., k_r) \in \{0,1\}^r; \sum k_i = s} (\cdots)$$

(formula (2.13))

• Leading term in De Concini–Kac filtration:

$$\operatorname{lt}(d_{r,s}) = c \cdot t_{1,1} t_{2,3} \ldots t_{r, r-s} \ldots t_{r, r}$$

(formula (3.1)), for the unique permutation whose height is

$\operatorname{ht}(\tau) = 2s(r-s)$

(formula (3.5)).

Conclusion

The Gelfand–Tsetlin subalgebra in the quantum enveloping algebra $U_q(\mathfrak{gl}_n)$, precisely characterized via explicit generators and their dominant terms with respect to the De Concini–Kac filtration, underpins the analysis of module spectra, algebraic independence, and the graded structure of quantum groups. The identification and computation of leading terms consolidate the connection between filtered algebra techniques, Galois order methods, and quantum generalizations of classical combinatorial representation theory, ensuring that the subalgebra occupies a central role in modern studies of quantum group representations (Futorny et al., 2012).