  The dual superconductor picture is one of the most promising scenarios for
quark confinement. To investigate this picture in a gauge-invariant manner, we
have proposed a new formulation of Yang-Mills theory, named the decomposition
method, on the lattice. The so-called restricted field obtained from the
gauge-covariant decomposition plays the dominant role in quark confinement. It
has been known by preceding works that the restricted-field dominance is not
observed for the Wilson loop in higher representations if the restricted part
of the Wilson loop is obtained by adopting the Abelian projection or the field
decomposition naively in the same way as done in the fundamental
representation. Recently, through the non-Abelian Stokes theorem (NAST) for the
Wilson loop operator, we have proposed suitable gauge-invariant operators
constructed from the restricted field to reproduce the correct behavior of the
original Wilson loop averages for higher representations. We have demonstrated
the numerical evidence for the restricted-field dominance in the string
tension.
  In this talk, we focus on the magnetic monopole. According to this picture,
magnetic monopoles causing the dual superconductivity are the dominant degrees
of freedom responsible for confinement. With the help of the NAST, we define
the magnetic monopole and the string tension extracted from the
magnetic-monopole part of the Wilson loop in a gauge-invariant manner. We will
further perform lattice simulations to measure the static potential for quarks
in higher representations using the proposed operators and examine the magnetic
monopole dominance in the string tension, which means that the string tension
extracted from the magnetic-monopole part of the Wilson loop reproduces the
proper string tension obtained from the original Wilson loop.
