The Lannister Ruling Structure During Tywin's Tenure as Hand of the King

When Tywin Lannister served as the Hand of the King under King Robert Baratheon, his son Jaime Lannister held the position of acting Lord of Casterly Rock. This arrangement was a clear demonstration of the complex relationships and responsibilities that existed within the Lannister family and the broader political landscape of Westeros.

Jaime Lannister's Role during Tywin's Tenure

As Tywin was the Hand of the King, his son Jaime Lannister would have been the acting Lord of Casterly Rock in his absence. This situation highlights the pragmatic administrative divide between the responsibilities of a Hand and a lord of a fortified seat. Although Tywin was managing the realm's affairs as Hand, Jaime was tasked with maintaining the familial rule over their ancestral home, ensuring the smooth running of the Lannister family's interests and land.

Casterly Rock and the Castellan

It is likely that Tywin had appointed a reliable and competent person to serve as the castellan of Casterly Rock during his tenure as Hand. This individual would have been responsible for the day-to-day running of the castle, making quick decisions and managing the various domestic and military affairs that arose during Tywin's absence.

The most probable candidate for this position is Kevan Lannister, Tywin's trusted right-hand man and a capable administrator. Kevan's position as Hand in Waiting or Secretary of State would have given him the necessary experience and authority to manage Casterly Rock effectively. Additionally, Kevan would have been well-prepared to handle the administrative responsibilities should Jaime be called away to fight in his campaign against the Kingswood Brotherhood.

The Unexpected Entrant: Cersei Lannister

During this period, another important Lannister, Cersei, was not at Casterly Rock. Cersei Lannister was residing in King's Landing with her father, Tywin, as he resigned his position as Hand. This situation neither allowed her to take charge of Casterly Rock nor assumed significant administrative roles in the west. Such an arrangement ensured that the responsibility and power were kept firmly within the Lannister family structure, with Kevan and Jaime handling the crucial affairs.

Robb Stark and Catelyn's Responsibilities at Winterfell

In stark contrast to the Lannister ruling structure, Robb Stark was not Lord of Winterfell when his father, Ned, was serving as Hand of the King. In Westeros, a lord does not automatically abdicate his position when serving as Hand. Instead, Robb, as Ned's heir, was involved in the day-to-day running of Winterfell.

However, Ned appointed Catelyn Stark to be the one in charge of Winterfell when he went to King's Landing. Due to a series of unfortunate events, Catelyn found herself mostly catatonic with grief over Bran's incident before being whisked away to King's Landing to warn Ned. Catelyn’s absence left the responsibility of Winterfell in the hands of other senior figures, including Maester Luwin. Later, Ser Rodrik Cassel, the master of arms and castellan, took over the day-to-day running of the castle while Robb was away, handling both the administrative and symbolic duties.

Summary

During Tywin Lannister's tenure as Hand of the King, the Lannister family maintained a clear and strategic division of responsibilities. While Kevan Lannister would have been the effective ruler at Casterly Rock, Jaime managed the day-to-day operations of the castle, and Cersei was not in the position to take charge. For the Starks, the seeming absence of Ned Stark as Hand meant that Robb was involved in running Winterfell, and Catelyn acted as the de facto leader, with other trusted individuals managing various aspects of the castle.