I am working with polynomial representations of spherical subalgebra of double affine Hecke algebra (DAHA) for $\mathfrak{gl}_n$. Let's call this algebra $\mathfrak{A}_n$ for short. Typically we think of DAHA as a $\mathbb{C}(q,t)$-algebra generated by $X_i, Y_i, T_i, i=1,\dots,n$ modulo certain relations. Then $\mathfrak{A}_n$ can be obtained by conjugation of each element of DAHA by the complete idempotent $S$.

It is known that there is a representation of $\mathfrak{A}_n$ in $\mathbb{C}[x_1,\dots,x_n]^{S_n}$ such that symmetric polynomials of $Y_i$ of DAHA act as Macdonald operators whose eigenvalues are Macdonald polynomials $P_{\lambda}(x_1,\dots,x_n;q,t)$ parameterized by Young diagrams $\lambda=\{\lambda_1,\dots\lambda_n\}$ with $n$ columns. In the basis of Macdonald polynomials one can find an explicit realization of all the other generators, in particular \begin{equation} Z^n_{0,l}= S\sum_i Y_i^l\, S\,,\quad Z^n_{0,-l}= q^l S\sum_i Y_i^l\, S\,,\quad Z^n_{l,0}= q^l S\sum_i X_i^l\, S\,,\quad Z^n_{-l,0}=S\sum_i X_i^l\, S\, \end{equation} It can be shown [Cherednik, Schiffmann-Vaserot] that the above elements generate entire $\mathfrak{A}_n$.

I am interested in explicit formulae in the above basis (or any other basis for that matter) which would allow me to construct raising (lowering) operators acting on such module which would add (remove) a single box to (from) Young tableaux $\lambda$.

At the moment I can only do it for $\mathfrak{sl}_2$ spherical DAHA. There must be a clever way to get the answer without explicitly symmetrizing things with the idempotent. Already in rank one case the sought raising and lowering operators look extremely simple in the symmetrized basis, while the corresponding intertwiners from Cherednik's book are more involved.

I am also aware of numerous constructions in elliptic Hall algebra (aka shuffle algebra), where people can compute some matrix elements between various Young diagrams, but that does not immediately provide an answer to my question despite proven, by Schiffmann and Vasserot, homomorphism between elliptic Hall and $\mathfrak{A}_n$.

  • 1
    $\begingroup$ The operators $Z_{1,l}$ act in the Macdonald basis as $Z_{1,l}\cdot P_\lambda = \sum_{\mu = \lambda + 1} a(i,\mu-\lambda) P_\mu$, where $\lambda + 1$ is the set of partitions obtained from $\lambda$ by adding one box, and the $a(i,\mu-\lambda)$ are some constants. I'm pretty sure the Schiffmann-Vasserot paper has an explicit expression in the DAHA for $Z_{1,l}$, it's one of their hard computational lemmas, I think. If you want some operator $F_i$ that acts on $P_\lambda$ by adding a box of content $i$, if possible, and by 0 otherwise, I'm not sure if this is inside the DAHA. $\endgroup$ – Peter Samuelson Mar 28 '18 at 8:22
  • $\begingroup$ Inside DAHA or spherical DAHA? $\endgroup$ – Peter Koroteev Mar 28 '18 at 21:57
  • $\begingroup$ I mean if such raising/lowering operators do not exist the corresponding matrix elements in shuffle algebra would vanish, and I don't think it is the case. $\endgroup$ – Peter Koroteev Mar 28 '18 at 22:34
  • $\begingroup$ Multiplying by $p_1$ is the operator $Z^1_{1,0}$ in the spherical DAHA, and when it acts on $s_\lambda$, it outputs the sum of all $\lambda'$ which are obtained from $\lambda$ by adding one box. Is this what you mean by a raising operator? Or do you want the raising operator to output just one $s_{\lambda'}$, and not a sum of them? $\endgroup$ – Peter Samuelson Apr 2 '18 at 16:03
  • $\begingroup$ Yes, I want to re-packages to operators so that $\lambda'$ differs from $\lambda$ by one additional box located at $(I,j)$. $\endgroup$ – Peter Koroteev Apr 27 '18 at 22:37

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