### Session III.6 - Symbolic Analysis

Monday, June 19, 15:30 ~ 16:00

## On Normal Forms in Differential Galois Theory for the Classical Groups

### Matthias Seiß

#### Universität Kassel, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

In classical Galois theory there is the well-known construction of the general polynomial equation over $\mathbb{Q}$ with Galois group the symmetric group $S_n$. Shortly recalling this construction we consider a rational function field $\mathbb{Q}(\boldsymbol{x})$ in $n$ indeterminates $\boldsymbol{x}=(x_1,\dots,x_n)$ over $\mathbb{Q}$ on which the symmetric group acts by permuting the variables $\boldsymbol{x}$. The fix field under this action is generated over $\mathbb{Q}$ by the elementary symmetric polynomials \[\boldsymbol{s}(\boldsymbol{x})=(s_1(\boldsymbol{x}),\dots,s_n(\boldsymbol{x}))\] and has transcendence degree $n$ over $\mathbb{Q}$. Moreover, $\mathbb{Q}(\boldsymbol{x})$ is a Galois extension of $\mathbb{Q}(\boldsymbol{s}(\boldsymbol{x}))$ with Galois group $S_n$ and its defining equation is the polynomial equation of degree $n$ whose coefficients are (up to sign) the polynomials $\boldsymbol{s}(\boldsymbol{x})$. Every algebraic extension of $\mathbb{Q}$ which is defined by a polynomial of degree $n$ is obtained as a specialization by substituting the roots for $\boldsymbol{x}$ in the general equation.

In differential Galois theory there is a similar construction of a general differential equation with differential Galois group the general linear group $\mathrm{GL}_n$. More precisely, for an algebraically closed field $C$ of characteristic zero one considers here a differential field $C\langle \boldsymbol{y} \rangle$ which is generated by $n$ differential indeterminates $\boldsymbol{y}=(y_1,\dots,y_n)$. Now the group $\mathrm{GL}_n(C)$ acts on $C\langle \boldsymbol{y} \rangle$ by linearly transforming the indeterminates $\boldsymbol{y}$. For a new differential indeterminate $Y$ the general differential equation is defined as the quotient of the two Wronskians \[ \frac{wr(Y,y_1,\dots,y_n)}{wr(y_1,\dots.y_n)} =: Y^{(n)} + c_{n-1}(\boldsymbol{y})Y^{(n-1)} + \dots + c_0(\boldsymbol{y}) Y=0. \] The coefficients $\boldsymbol{c}=(c_{n-1}(\boldsymbol{y}),\dots,c_0(\boldsymbol{y}))$ of the general equation are differentially algebraically independent over $C$ and as the elementary symmetric polynomials they generate the fixed field of $C\langle \boldsymbol{y} \rangle$ under $\mathrm{GL}_n(C)$ over the constants. Clearly, $C\langle \boldsymbol{y} \rangle$ is a Picard-Vessiot extension of $C\langle \boldsymbol{c} \rangle$ with differential Galois group $\mathrm{GL}_n(C)$ and it is generic extension. Indeed, let $E$ be a Picard-Vessiot extension of a differential field $F$ with constants $C$ defined by a linear scalar differential equation of order $n$. If $\boldsymbol{\eta}=(\eta_1,\dots,\eta_n)$ are the linearly independent solutions of this equation, then $E/F$ is obtained as a specialization of the general equation by substituting the solutions $\boldsymbol{\eta}$ for the indeterminates $\boldsymbol{y}$. Generalizations to groups other than $\mathrm{GL}_n(C)$ were obtained in [1] and [2]. In all these cases the general differential equation involves $n$ differential indeterminates over $C$ (apart from $Y$).

An analogue construction of a general linear differential equation for the classical groups was presented in [5]. This approach combines the geometric structure of a classical group $G(C)$ of Lie rank $l$ with Picard-Vessiot theory. As in the case of the general equation with group $\mathrm{GL}_n(C)$ the construction starts with a differential field $C\langle \boldsymbol{v} \rangle$ generated by $l$ differential indeterminates $\boldsymbol{v}=(v_1,\dots,v_l)$ over $C$, but this time the general extension field is a Liouvillian extension $\mathcal{E}$ of $C\langle \boldsymbol{v} \rangle$ with differential Galois group a fixed Borel group $B^-(C)$ of $G(C)$. Choosing a Chevalley basis of the Lie algebra $\mathfrak{g}(C)$ of $G(C)$ the Liouvillian extension is defined by a matrix $A_{\mathrm{Liou}}(\boldsymbol{v})$ which is the sum of the Cartan subalgebra parametrized by the indeterminates $\boldsymbol{v}$ and the basis elements of the root spaces corresponding to the negative simple roots. In order to define a group action of $G(C)$ on $\mathcal{E}$ one needs to construct a fundamental matrix $\mathcal{Y}$ and let $G(C)$ act on it by right multiplication which will then induce an action of $G(C)$ on $\mathcal{E}$. To this end, let $b$ be a fundamental matrix of $A_{\mathrm{Liou}}(\boldsymbol{v})$ in $B^-(\mathcal{E})$, $\overline{w}$ a representative of the longest Weyl group element and let $u(\boldsymbol{v}, \boldsymbol{f})$ be the product of matrices of all negative root groups where the matrices corresponding to the negative simple roots are parametrized by $\boldsymbol{v}$ and the matrices corresponding to all remaining negative roots depend on differential polynomials $\boldsymbol{f}$ in $C\{ \boldsymbol{v}\}$. These differential polynomials are chosen in such a way that the logarithmic derivative of $\mathcal{Y} = u \overline{w} b$ is the matrix $A_G(\boldsymbol{s}(\boldsymbol{v}))$ constructed in [4] and [5], where $\boldsymbol{s}(\boldsymbol{v})$ are $l$ differential polynomials in $C\{ \boldsymbol{v}\}$. Analogue to the cases of the symmetric group and $\mathrm{GL}_n(C)$ presented above, the $\boldsymbol{s}(\boldsymbol{v})$ are differentially algebraically independent over $C$. Multiplying $\mathcal{Y}$ from the right with elements of $G(C)$ and then recomputing the Bruhat decomposition of the product defines an action on $\boldsymbol{v}$, $\boldsymbol{f}$ and on the generators of the Liouvillian extension, i.e. the entries of $b$, and so induces an action of $G(C)$ on $\mathcal{E}$. The fixed field under this induced action is $C\langle \boldsymbol{s}(\boldsymbol{v}) \rangle$ and one can show that the field $\mathcal{E}$ is a Picard-Vessiot extension of $C\langle \boldsymbol{s}(\boldsymbol{v}) \rangle$ with differential Galois group $G(C)$ for the differential equation defined by $ A_G(\boldsymbol{s}(\boldsymbol{v}))$. The construction is only generic for Picard-Vessiot extensions of $F$ with defining matrix gauge equivalent to a matrix in \emph{normal form}, i.e.\ a specialization of $A_{G}(\boldsymbol{s}(\boldsymbol{v}))$. Deciding such a gauge equivalence is non-trivial as a consequence of the fact that $\mathcal{E}$ and $C\langle \boldsymbol{s}(\boldsymbol{v}) \rangle $ have differential transcendence degree $l$ over $C$.

This talk is dedicated to the question of the genericity properties of the extension $\mathcal{E}$ over $C\langle \boldsymbol{s}(\boldsymbol{v}) \rangle$. We consider the problem of gauge equivalence of a generic element of the Lie algebra to a matrix in normal form. More precisely, let $d$ be the dimension of the classical group $G$ and let $\boldsymbol{a}=(a_1,\dots,a_d)$ be differential indeterminates over a differential field $F$ with constants $C$. Further let $A(\boldsymbol{a})$ be a generic element in the Lie algebra $ \mathfrak{g}(F\langle \boldsymbol{a} \rangle)$ obtained from parametrizing the Chevalley basis from above with the indeterminates $\boldsymbol{a}$. It is known (cf. [3]) that the differential Galois group of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $F \langle \boldsymbol{a} \rangle$ is $G(C)$. We present the construction of a differential field extension $\mathcal{L}$ of $F \langle \boldsymbol{a} \rangle$ such that the field of constants of $\mathcal{L}$ is $C$, the differential Galois group of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $\mathcal{L}$ is still the full group $G(C)$ and $A(\boldsymbol{a})$ is gauge equivalent over $\mathcal{L}$ to a specialization of $A_G(\boldsymbol{s}(\boldsymbol{v}))$, i.e. to a matrix in normal. In the special case of $G=\mathrm{SL}_3$ we show how one obtains an analogous result for specializations of the coefficients of $A(\boldsymbol{a})$.

$\textbf{References}$

[1] L. Goldman, Specialization and Picard-Vessiot theory. Transactions of the American Mathematical Society, 85:327–356, 1957.

[2] L. Juan and A. Magid, Generic rings for Picard-Vessiot extensions and generic differential equations. Journal of Pure and Applied Algebra, 209(3):793–800, 2007.

[3] L. Juan, Pure Picard-Vessiot extensions with generic properties. Proceedings of the American Mathematical Society, 132(9):2549–2556, 2004.

[4] M. Seiss, Root Parametrized Differential Equations for the Classical Groups. https://arxiv.org/abs/1609.05535.

[5] M. Seiss, On the Generic Inverse Problem for the Classical Groups. https://arxiv.org/abs/2008.12081.

Joint work with Daniel Robertz (RWTH Aachen University, Germany).